Chapter 2: Methods of Psychology
In 1960, a young researcher named Jane Goodall arrived at the Gombe Stream reserve in Tanzania with little more than a notebook and a pair of binoculars. She had no laboratory and no equipment for running tests. Her plan was simple but demanding: to sit quietly, day after day, and watch wild chimpanzees go about their lives.
For weeks the animals fled whenever she came near. Slowly they grew used to her presence, and she began to record what they actually did — how they groomed one another, quarrelled, shared food and raised their young. She interfered with nothing. She only watched, patiently and systematically.
One morning she saw a chimpanzee strip the leaves from a twig, push it into a termite mound, and draw out insects to eat. The animal had made a tool. Until then, many scientists believed that only human beings could make and use tools. A single careful observation had overturned a long-held belief.
Goodall’s work points to something at the heart of psychology. Behaviour can be studied without disturbing it, simply by watching closely and keeping honest records. But observation is only one of the methods a psychologist can use, and it cannot answer every question. This chapter introduces the different ways psychologists study behaviour, and the tools they use to make sense of what they find.
Diagram
Fig. 2.0-1 – illustration of a lone researcher seated with a notebook, quietly observing a small group of animals or people in a natural outdoor setting at a distance, representing naturalistic observation
By the end of this chapter you will be able to answer each of these questions for yourself.
2.1 The Scientific Study of Behaviour
Psychology is often defined in a single line: it is the scientific study of behaviour. The word that does the real work in that sentence is scientific. It separates the way a psychologist studies people from the way the rest of us do it every day.
All of us form opinions about why people act as they do. Your friend is quiet today, so you decide she is upset. But such everyday judgements are quick, private and shaped by our own moods and biases. They are easy to make and easy to get wrong.
Psychology sets a higher standard. It studies behaviour using methods that are planned in advance, recorded carefully, and open to checking by others. This is what turns loose impressions into knowledge.
What Makes a Method Scientific
A scientific method is an organised, step-by-step way of investigating a question so that other people can examine the work and repeat it. Three features give a method this scientific character.
First, it is empirical, meaning it is based on evidence actually gathered through observation or measurement, rather than on opinion or common sense. Second, it has objectivity, meaning the findings are kept free from the personal bias, wishes and expectations of the investigator. Third, it has replicability, meaning that another researcher who follows the same steps should arrive at similar results.
From Looking Within to Looking Outward
The earliest method psychologists used fell short of these standards. In the late nineteenth century, when psychology was first taking shape as a separate science, researchers relied on introspection — literally “looking within.” Trained participants were asked to observe their own thoughts, sensations and feelings and to report them as accurately as they could.
Introspection had a serious weakness. A report of one’s private inner experience cannot be checked by anyone else. Two people looking within might describe the same sensation quite differently, and no outsider could decide who was correct. Because it lacked objectivity, introspection could not settle disagreements, and its use steadily declined.
This weakness pushed psychology towards methods that study behaviour from the outside — methods in which the evidence is open to view and can be recorded, measured and repeated. Behaviour became the focus precisely because it can be observed by more than one person.
The Methods Ahead
This chapter surveys the main scientific methods psychologists use to study behaviour: observation, the case study, the survey, the interview and the experiment. Alongside these it looks at how researchers select whom to study through sampling, how psychological tests measure mental qualities, and the ethical rules that must guide any study of human beings.
The chapter then turns to statistics — the set of tools that help a psychologist organise the scores a study produces and draw sound conclusions from them. As shown in Fig. 2.1-1, the methods and the statistical tools together form the working equipment of the discipline.
Diagram
Fig. 2.1-1 – overview chart with “Methods of Psychology” at the centre, branching into two groups: (1) ways of collecting data — observation, case study, survey, interview, experiment; and (2) ways of selecting and interpreting data — sampling, psychological tests, statistics
No single method is best for every question. Each has strengths that suit it to some problems and weaknesses that rule it out for others. Learning what each method can and cannot do is the real aim of this chapter.
2.2 Observational Method
You watch other people all the time. You notice who looks nervous before an exam, who dominates a group discussion, who hangs back at a party. Observation turns this ordinary act of watching into a scientific method by making it careful, planned and honest.
As a method of psychology, observation means recording, perceiving, analysing and interpreting other people’s behaviour by watching it closely. It rests on a simple assumption: that behaviour we can see is the outward sign of mental activity we cannot see.
Psychologists call the behaviour we can directly observe overt behaviour — a smile, a raised voice, a hand reaching out. The thoughts and feelings hidden behind it are covert behaviour, which no observer can see directly. Observation works with the overt in order to draw careful inferences about the covert.
Good observation is not casual glancing. The observer decides in advance what to watch for, plans how to record it, fixes a sensible time limit, and chooses a suitable number of people — enough to be meaningful, not so many that watching becomes impossible. The observer must also be trained to notice the important behaviours and ignore the trivial ones.
Depending on how much the situation is controlled, observation takes different forms. Fig. 2.2-1 shows the main types. We begin with the two that differ most in their setting: naturalistic and scientific observation.
Diagram
Fig. 2.2-1 – a simple branching chart of the “Types of Observation”: naturalistic observation, scientific observation (field study), longitudinal method, and cross-sectional method
2.2.1 Naturalistic Observation
In naturalistic observation, behaviour is watched in its natural, real-life setting, exactly as it happens, without the observer interfering in any way. Because nothing is arranged or controlled, it is also called uncontrolled or unstructured observation.
Think of a psychologist sitting quietly at the edge of a school playground, noting how children form groups, share toys and settle quarrels during the lunch break. The children behave as they always do, because the situation has not been altered for them.
This is exactly the method you met in the chapter’s opening story, where a researcher watched wild chimpanzees for months without disturbing them. The great strength of the method comes from this hands-off quality.
The main advantage of naturalistic observation is its high ecological validity, which means the findings genuinely reflect how people behave in real life rather than in an artificial setup. Because the behaviour is spontaneous, what the observer records is true to everyday life.
Its main limitation is that the observer controls nothing. Many influences act on the behaviour at once, so it is very difficult to say what actually caused it. The method can describe behaviour faithfully, but it cannot establish cause and effect.
2.2.2 Scientific Observation (Field Study)
Sometimes a psychologist wants to do more than watch — they want to see what happens when one factor in a real-life situation is changed. In scientific observation, certain factors are deliberately controlled or manipulated, and their effect on behaviour is observed. Because it takes place in a real setting such as a school, market or workplace, it is also called a field study, and because the situation is arranged, it is a controlled or structured form of observation.
A field study does not begin with idle watching. It starts from a clearly defined problem and a hypothesis — a testable prediction about what the researcher expects to find. The observation is then planned to test that prediction.
For example, a researcher might study whether background music changes how long shoppers stay in a store, playing music on some days and none on others while recording the time customers spend inside.
The main advantage of scientific observation is that it is planned and controlled, so a cause-and-effect relationship can be examined far more closely than in naturalistic observation, and the study can be repeated by others.
Its main limitation is that real settings contain countless other influences that are hard to control. Because these cannot all be eliminated, a clean link between the chosen factor and the behaviour may still be missed.
2.2.3 Longitudinal and Cross-Sectional Methods
Two further methods are observational in nature but built to answer a special kind of question: how does behaviour change as people grow older? Both are widely used in developmental psychology to trace developmental trends across time. They differ in a simple but important way — whether they follow the same people or compare different ones.
2.2.3.1 Longitudinal Method
The longitudinal method studies the same group of people repeatedly, at different time intervals, over a long period. The researcher watches how a particular trait changes in these individuals as the years pass.
Suppose a psychologist follows the same forty children from toddlerhood to adolescence, recording how their friendships change across fifteen years. Because it is the same children throughout, the changes seen can be put down to development itself, and not to differences between one set of children and another.
The main advantage of the longitudinal method is that, since the very same people are followed, changes over time can be traced without confusing them with differences between different individuals. This makes it strong for studying the causes of developmental change.
Its main limitation is that it is extremely lengthy and expensive, and participants tend to drop out over the years through relocation or loss of interest, shrinking the group by the study’s end.
2.2.3.2 Cross-Sectional Method
The cross-sectional method takes the opposite approach. Instead of following one group over many years, it studies several different groups, differing in one feature such as age but similar in others, all at a single point in time.
To study how memory changes with age, a psychologist using this method would test groups of twenty-, forty- and sixty-year-olds this month and compare their scores, rather than waiting forty years to test one group repeatedly.
The main advantage of the cross-sectional method is that it is quick and economical, since all the data are gathered once and several age groups can be compared at the same time.
Its main limitation is that it gives only a snapshot. Differences between the age groups may arise because they grew up in different times, not because of development itself, so a clear cause-and-effect account of change cannot be given. Fig. 2.2-2 contrasts the two designs at a glance.
Diagram
Fig. 2.2-2 – side-by-side comparison: on the left, one group of people shown ageing across several time points (longitudinal); on the right, several groups of different ages shown at one single time point (cross-sectional)
2.3 Case Study Method
Some questions in psychology cannot be answered by watching many people briefly. They need a long, deep look at one person. Why did a particular child develop a fear of dogs? How does one patient recover after a brain injury? For questions like these, psychologists use the case study method, also known as the case history method.
A case study is an in-depth investigation of a single individual, group or event. Instead of collecting a little information from many people, it collects a great deal of information about one case, building a detailed picture over time.
The word “case” does not always mean one person. A case can be a family, a small social group or even a single organisation. A study of how one growing company handles its staff would also count as a case study.
Where the Information Comes From
A case study gathers information from as many sources as possible. The researcher studies the person’s background in detail — birth, childhood, education, family life, work history and important relationships — along with their ambitions, likes, dislikes and attitudes.
To check and add to this account, the researcher also speaks to family members, close relatives and friends, and draws on records such as diaries, school reports, clinical notes, letters and photographs. All of it is then analysed together to build a rounded profile, as shown in Fig. 2.3-1.
Diagram
Fig. 2.3-1 – several data sources (interviews, observation, psychological tests, personal records such as diaries and school reports, and accounts from family and friends) shown as arrows converging on a central “case profile”
The case study is widely used in medicine, psychiatry and abnormal psychology. Much of Sigmund Freud’s work in building his theories grew out of detailed case studies of his patients.
The main advantage of the case study method is the depth it offers. It captures rich, detailed information about one case that broader methods would miss, and it can be used to study rare or unusual situations that could never be created in an experiment.
Its main limitation is that the findings cannot be generalised. Because only one case, or a few, is studied, that case may not represent other people, so what is true of it may not be true of anyone else.
2.4 Survey Method
Suppose you wanted to know whether teenagers in India prefer shopping online or in stores. You could not sit and study each teenager in depth — there are far too many. What you need is a way to collect a little information from a great many people. This is what the survey method does.
A survey is a technique for studying a whole population, or a group drawn from it, with respect to some psychological or social variable such as an opinion, attitude or preference. Information is gathered from many people by asking them a series of questions, through a questionnaire, a self-report or an interview, and by post, telephone or in person.
In practice, a researcher rarely questions every single person in a population. Instead they study a sample survey — a smaller group chosen to represent the whole — and then extend the findings to the population. You will see how such samples are chosen later in this chapter.
Surveys are the standard tool for measuring public opinion: what people think of a product, a television programme or a social issue, and how they may vote in an election.
For a survey’s results to be trusted, it must be built carefully. This is why surveys are put through standardisation — the questions, the way they are asked, and the way answers are scored are all kept uniform for every respondent. Only then can the results be reliable and fairly compared.
The main advantage of the survey method is its reach: it can collect a large amount of information from very large samples, or even whole populations, in a relatively short time.
Its main limitation is depth. A survey gives a broad, surface-level picture of what people think, but it cannot explore any one person’s reasons and feelings in the way a case study can.
2.4.1 Questionnaire
A questionnaire is the survey’s most common instrument. It is a written or printed set of well-structured questions on a particular topic, given to an individual or a group so that their answers can be collected and studied.
A good questionnaire is not thrown together. Its questions must be clear and easy to understand, worded in the same way for everyone so they cannot be read two different ways, engaging enough that respondents do not lose interest, and short and to the point. A confusing or tiring questionnaire produces unreliable answers.
The questions themselves come in two forms. A closed-ended question offers fixed options to choose from, while an open-ended question lets people answer freely in their own words. Each has its place, as the comparison below shows.
2.4.2 Inventory
An inventory, or self-report inventory, is a close relative of the questionnaire that is widely used in psychological testing. It presents an organised set of statements, each with a fixed range of response choices, and the respondent selects the option that best fits their own opinion or feeling, as shown in Fig. 2.4-1.
Diagram
Fig. 2.4-1 – a sample inventory item showing a statement such as “I enjoy meeting new people,” followed by a five-point response scale ranging from “Strongly Disagree” to “Strongly Agree”
Inventories are commonly used to measure interests, values and personality traits. Well-known examples include the Minnesota Multiphasic Personality Inventory, Bell’s Adjustment Inventory and the California Psychological Inventory.
Questionnaires and inventories share a clear strength: they are quick and economical, since one form can be given to a large number of people at once, even by post, without the researcher being present.
They also share one important weakness — respondents may not answer honestly. To appear in a good light, people often give the answer they think is expected of them rather than the truth. This tendency is called social desirability, and it can quietly distort the results of any self-report.
2.5 Interview Method
Almost everyone faces an interview at some point — for a school admission, a scholarship or a job. The interview method is one of the oldest and most direct ways of studying a person, and psychologists use it as a research tool in much the same way.
An interview is a face-to-face interaction between two people. One, the interviewer, asks a set of verbal questions; the other, the interviewee, answers them. Through this exchange the interviewer collects information about the interviewee’s opinions, attitudes and thoughts.
An interview offers something a written form cannot. Besides the spoken answers, the interviewer can watch the interviewee’s non-verbal cues — body language, eye contact, facial expressions and reactions to particular questions. Read carefully, these cues add valuable meaning to what is actually said, as Fig. 2.5-1 suggests.
Diagram
Fig. 2.5-1 – two people seated facing each other in conversation; a small note beside the interviewer points to the interviewee’s posture, eye contact and expression, labelled “non-verbal cues”
To keep an interview fair and comparable, interviewers often use an interview schedule — a fixed set of questions asked in the same order to every interviewee. This standardised form makes it easier to compare one person’s answers with another’s.
Students sometimes confuse an interview schedule with a questionnaire, since both are lists of questions.
The interview is widely used to select people for jobs and professional courses. It is also central to psychotherapeutic and medical work, where it helps diagnose and treat problems, and it is used in legal questioning.
The main advantage of the interview method is that, being face-to-face and flexible, it lets the interviewer adapt questions to the situation and read non-verbal cues — drawing out information that a fixed written form could never capture.
Its main limitation is that the outcome depends heavily on the interviewer. A skilled interviewer is not always available, and the interviewer’s own bias may shape which questions are asked and how the answers are understood.
2.6 Experimental Method
The methods so far can tell us what happens and how often, but they struggle with the hardest question of all: what actually causes a behaviour. The experimental method was built to answer exactly that. It is the pillar of scientific inquiry, and it is the only method that can firmly establish a cause-and-effect relationship.
The idea behind it is simple. The experimenter deliberately changes one factor, holds everything else steady, and watches what happens. If behaviour changes, it can be traced to the one factor that was changed. An experiment therefore involves manipulating a cause, measuring its effect, and controlling everything else that might interfere.
2.6.1 Variables in an Experiment
A variable is any characteristic, event or condition that can change, and that can be manipulated, controlled or measured in a study. An experiment is really a careful arrangement of variables.
Three kinds of variable appear in every experiment. The independent variable (IV) is the factor the experimenter deliberately changes — the suspected cause. Because it is the stimulus that sets off a response, it is also called the stimulus variable.
The dependent variable (DV) is the behaviour the experimenter measures — the effect. Its value is expected to depend on the independent variable, which is why it is also called the response variable.
The extraneous variables are all the other factors that could also affect the dependent variable. These are not the focus of the study, but they must be controlled, or they will muddy the results.
Consider an experiment on whether brighter light helps reading. The experimenter varies the light and counts how many pages a person reads in a fixed time. Here the light intensity is the independent variable, the number of pages read is the dependent variable, and factors such as the reader’s eyesight, tiredness, posture and noise in the room are extraneous variables, as illustrated in Fig. 2.6-1.
Diagram
Fig. 2.6-1 – the independent variable (cause) shown with an arrow leading to the dependent variable (effect); a set of extraneous/confounding variables shown below, with arrows pointing at both, indicating the influences the experimenter must control
When extraneous variables are not properly controlled, they become confounding variables — hidden influences that distort the results, so we can no longer tell whether the independent variable or something else produced the effect. One common example is experimenter bias, where the experimenter’s own expectations are unknowingly passed on to the participants, nudging them to behave as expected.
2.6.2 Control: Experimental and Control Situations
To be sure the independent variable is responsible for a change, the experimenter compares two situations. In the experimental group, the independent variable is present. In the control group, it is absent, and behaviour is measured under normal conditions.
The two groups are kept alike in every important respect except one — the presence of the independent variable. Any difference in their behaviour can then be put down to that single difference. This is why the control group is so valuable: it is the benchmark against which the effect of the independent variable is judged.
For example, to test whether a new memory technique improves recall, a researcher teaches the technique to the experimental group but not to the control group, then tests both. If the experimental group recalls more, the technique gets the credit — because that is the only way the two groups differed.
2.6.3 Steps in an Experiment
An experiment follows an orderly sequence, where each step prepares the way for the next. Fig. 2.6-2 sets out the full path.
The researcher first identifies a problem or question that interests them. From a broader theory — a general explanation of some behaviour — they form a hypothesis, the specific, testable prediction introduced earlier in this chapter.
They then design the experiment: identifying the independent and dependent variables, and identifying and controlling the extraneous variables. Next they select a sample from the population and assign its members to the experimental and control conditions.
The data are then collected and treated statistically to organise them and draw out what they show. On this basis the hypothesis is either accepted or rejected — accepted if the results match the prediction, rejected if they do not — and a conclusion is drawn. Finally, the experiment is repeated some time later to check that it reliably yields similar results.
Diagram
Fig. 2.6-2 – flow: identify problem → form hypothesis → design experiment and identify variables → select and assign sample to groups → collect data → analyse data statistically → accept or reject hypothesis → draw conclusion → replicate
Strengths and Limits of the Experiment
The main advantage of the experimental method is that, because the variables are controlled, it alone can establish a clear cause-and-effect relationship. This makes it the most objective and repeatable of all the methods.
Its main limitation is that the controlled setting can be artificial. People may not behave in a laboratory as they would in real life, so experiments can have low ecological validity, and many real-life situations cannot be recreated in an experiment at all.
2.7 Sampling and Population
No researcher can study every person a conclusion is meant to apply to. If you want to know how Indian teenagers use social media, you cannot question all of them — there are tens of millions. So psychologists study a small group and carefully extend their findings to the larger one. Understanding how this works begins with two terms.
The population is the entire group the researcher wants to draw conclusions about — more precisely, the target population of the study. The sample is the smaller group, drawn from that population, which is actually studied. The process of selecting the sample is called sampling.
The results found in the sample are then generalised back to the population. As Fig. 2.7-1 shows, the sample is a manageable slice of a much larger whole.
Diagram
Fig. 2.7-1 – a large oval labelled “target population” filled with many figures, with a small circle of a few figures drawn out from it and labelled “sample,” an arrow linking the sample back to the population
The whole strategy depends on one thing: choosing the sample well. Psychologists distinguish three kinds of sample.
A representative sample is a small group that reflects all the important characteristics of the population, and in the same proportions. Because it mirrors the whole, conclusions drawn from it can safely be applied to the population.
A random sample is chosen so that every single member of the population has an equal chance of being selected, and the choice of one member does not affect the choice of another. This is the scientific way of selecting a sample, and a good random sample usually turns out to be representative.
A biased sample is chosen for the researcher’s convenience, or deliberately to make a point. Because its members are not selected fairly, it fails to reflect the population, and conclusions drawn from it can be badly mistaken.
These same three terms — statistics, sample and population — return when we turn to the statistical side of research later in the chapter, where they are treated as the basic building blocks of data analysis.
2.8 Psychological Tests
How do you measure something you cannot see, such as a person’s intelligence, anxiety or interest in music? Psychologists do it with a psychological test — a standardised, systematic tool for measuring some characteristic of behaviour.
A psychological test can measure behaviour that is cognitive (to do with thinking and knowing), conative (to do with willing and doing) or affective (to do with feeling and emotion), and it can do so in numbers or in qualities.
Three ideas sit inside this definition. A good test is objective, meaning it is free from the tester’s personal bias. It is standardised, following the uniform procedure introduced earlier when we discussed surveys. And it works from a sample of behaviour — a set of responses from which broader conclusions can be drawn.
2.8.1 Characteristics of a Psychological Test
A test is only worth using if it has certain qualities. Four matter most.
The first is reliability — a good test gives consistent results, producing similar scores when the same person is tested again under similar conditions. The second is validity — the test must actually measure what it claims to measure, and not something else.
The third quality is objectivity, meaning the test is administered and scored in the same way for everyone, leaving no room for the tester’s opinion to creep in. The fourth is having proper norms — established reference values, gathered from large groups, against which an individual’s score can be compared to see whether it is high, average or low.
2.8.2 Types of Psychological Tests
Psychological tests can be sorted in several ways, depending on what one looks at, as summarised in Fig. 2.8-1.
Diagram
Fig. 2.8-1 – a central label “Types of Psychological Tests” branching into four bases of classification: psychological trait measured, nature (individual/group), time taken (speed/power), and use of language (verbal/non-verbal/performance)
By the Trait Measured
An intelligence test measures a person’s intelligence, an aptitude test measures their potential to do well in a particular field, and a personality test identifies their personality traits, often using techniques such as interviews and case histories.
By Nature and by Time
By nature, a test is either an individual test, given to one person at a time, or a group test, given to many people together. By the time allowed, a speed test is strictly time-bound, while a power test has no time limit but arranges its items from easy to increasingly difficult.
By Use of Language
A verbal test requires the use of language to answer, while a non-verbal test uses pictures instead and suits people who are very young, unlettered, or from a different language background. A performance test asks the person to actually do a task, such as arranging blocks or solving a puzzle.
2.8.3 Uses of Psychological Tests
Psychological tests are put to many uses. Most basically, they detect and measure abilities, aptitudes, interests, personality and intelligence, and they allow us to measure individual differences — comparing one person with another, or the same person at different times.
In clinical settings, tests help professionals assess mental health conditions and judge their severity, which is why they are used in hospitals, counselling centres and rehabilitation work. It is important to remember, though, that a diagnosis is made by a trained professional using many sources of information — never from a single test score, a checklist, or an online quiz.
Tests also help classify people for educational and clinical purposes, build up a personality profile of an individual, provide valuable data for research, and guide the selection and placement of people in jobs, including specialised fields such as the armed forces.
2.9 Ethical Standards in Psychological Research
Because psychology studies real people — their thoughts, feelings and private lives — the researcher carries a serious responsibility to protect them. The rules that guide this are called ethical standards: the principles that govern what a researcher may and may not do to those taking part in a study.
Professional bodies such as the British Psychological Society and the American Psychological Association publish detailed codes of research ethics. The ISC syllabus highlights five standards in particular.
Voluntary Participation
Voluntary participation means people take part in a study of their own free will. No one may be pushed into it by pressure, force or manipulation, and a participant is free to withdraw at any stage, without penalty, even after the study has begun.
Informed Consent
Informed consent means that, before agreeing to take part, a person is given full information about the study — its purpose, what they will be asked to do, how long it will take, and any risks or benefits. Only then can their agreement be genuine. Where participants are children or otherwise unable to consent for themselves, permission must come from a parent or guardian.
Debriefing
Debriefing takes place after the study is over. The researcher explains its true purpose — especially if anything had to be held back during the study — answers the participant’s questions, and makes sure the person leaves without any lingering worry or discomfort.
Confidentiality of Data
Confidentiality of data means all personal information collected must be kept private and secure. Researchers use methods such as coding and anonymisation so that individuals cannot be identified, share the data only within the research team, and use it strictly for the stated purpose. A breach of confidentiality can cause real harm and destroys the trust on which research depends.
2.10 Use of Statistics in Interpreting Research Results
A study does not end when the data are collected. Imagine a psychologist facing a page of two hundred exam scores. On their own, those numbers say almost nothing — they are scattered and hard to grasp. To turn them into meaning, the psychologist uses statistics.
Statistics is the branch of mathematics that deals with collecting, organising, analysing and interpreting numerical data. In psychology it is the tool that reduces a mass of scores to clear conclusions, and it gives the subject much of its claim to be a science.
The statistician Henry Garrett described statistics as the science that handles complex numerical data — data shaped by many causes — and the collection and classification of numerical facts about large groups. Psychology depends on it to make its findings reliable and comparable.
2.10.1 Descriptive and Inferential Statistics
Statistical methods come in two broad kinds. Descriptive statistics summarise and describe a set of data in a meaningful way, so that its patterns can be seen. They stay within the data actually collected and do not reach beyond it.
Inferential statistics go further. They allow a researcher to generalise from a sample to the wider population, to estimate how strongly variables are related, and above all to judge whether a difference between groups is a real effect or just the result of chance.
2.10.2 Why Statistics is Used in Psychology
Statistics serves psychology in four main ways, shown together in Fig. 2.10-1.
First, it helps in describing and summarising data. A long list of scores can be reduced to a table, a graph or a single average, giving a clear overview that the raw numbers hide.
Second, it allows the comparing of individuals or groups. By working out the average score of two or more groups, a researcher can see, for example, whether one teaching method produced better results than another.
Third, it helps in investigating relationships between variables. Through a correlation — a measure of how two variables move together — statistics can show whether, and how strongly, two things such as study time and marks are related.
Fourth, it aids predicting. Because a study tests a hypothesis, statistics helps judge how likely a predicted outcome is, and whether the hypothesis should be accepted or rejected.
Diagram
Fig. 2.10-1 – a central label “Role of Statistics” branching into four uses: describing and summarising data, comparing individuals or groups, investigating relationships between variables, and predicting
2.10.3 Basic Statistical Concepts: Statistics, Sample, Population
Three terms form the vocabulary of any statistical study. The first, statistics, we have just met — it refers both to the methods used and to the numerical measures produced. A single such value calculated from a sample, such as its average score, is called a statistic.
The other two terms are already familiar from our discussion of sampling. A sample is the smaller group of people actually studied, and a population is the whole group the researcher wants the findings to apply to.
The link between them is the heart of statistics in psychology. A researcher computes a statistic from a sample and then uses inferential methods to decide how far it can be trusted as a description of the whole population.
2.11 Organising Scores into Frequency Distributions
Numbers straight out of a study are a jumble. Before a psychologist can find an average, spot a pattern or draw a conclusion, the scores must be tidied into an organised form. This section shows how raw scores are grouped so that they begin to make sense.
2.11.1 Data: Primary and Secondary
The word data means a collection of numerical facts, scores or given information. It is the plural of the Latin word datum. Each single score in a set — one student’s marks, say — is called an observation, and a set of such observations is the data.
Data come in two kinds according to who collected them. Primary data are collected first-hand by the researcher, for their own specific purpose. Gathering primary data takes time, but it is reliable and exactly suited to the study’s needs.
Secondary data are collected earlier by someone else and then reused by the present researcher, drawn from sources such as books, journals or websites. They save time but were gathered for a different purpose.
2.11.2 Ways of Organising Raw Data
Scores in their crude, unorganised form are called raw data or ungrouped data. They are scattered and reveal little until they are arranged. There are three common ways to organise them.
The first is rank order — arranging the scores in ascending or descending sequence, usually placing the highest at the top and the lowest at the bottom. This shows each score’s position at a glance.
The second is the statistical table — setting the scores out in rows and columns under clear headings, so that different categories can be compared.
The third, and the most powerful for large sets, is the frequency distribution. Here the number of times a score occurs is called its frequency, and scores are grouped so we can see how often each value, or each band of values, appears.
2.11.3 Constructing a Frequency Distribution Table
A frequency distribution table groups scores into bands and records how many scores fall into each band. Building one follows a few clear steps, and along the way a few terms are needed.
The range is the difference between the highest and the lowest score. A class interval is one of the bands into which scores are grouped, such as 30–34. The lower and upper values of a class are its class limits, and the value halfway across it — found by averaging its two limits — is the class midpoint (also called the class mark). Finally, the cumulative frequency of a class is the running total of all frequencies up to and including that class; it will be needed later when we find the median.
The steps for building the table are as follows. First, find the range. Second, decide the number of class intervals — a common rule of thumb is between 5 and 15. Third, work out the width of each interval using
\[i = \frac{\text{Range}}{\text{Number of class intervals}}\]
where \(i\) is the width, rounded up to a convenient whole number. Fourth, list the class intervals, tally each score into its class, and count the tallies to get the frequency of each class. The sum of all frequencies equals \(N\), the total number of scores.
2.12 Measures of Central Tendency
Once scores are organised, the next task is to describe them with a single, representative number. If a friend asks how your class did in a test, you would not read out forty marks — you would give one figure that stands for the whole group. Such a figure is a measure of central tendency.
A measure of central tendency is a single value that represents a whole set of scores by locating its centre — the value around which the scores tend to cluster. There are three such measures: the mean, the median and the mode.
2.12.1 Mean
The mean is the arithmetic average, and it is the most widely used and dependable measure of central tendency. It is found by adding up all the scores and dividing by how many there are. It is written as \(M\).
For ungrouped (raw) data, the formula is
\[M = \frac{\sum X}{N}\]
where \(X\) is each score, \(\sum X\) is the sum of all the scores, and \(N\) is the number of scores.
When data have been grouped into a frequency distribution, we use the midpoint of each class to stand for all the scores in it. The formula becomes
\[M = \frac{\sum fX}{N}\]
where \(X\) is the class midpoint, \(f\) is the frequency of the class, and \(N = \sum f\).
2.12.2 Median
The median is the middle score when all the scores are arranged in order. It divides the distribution into two equal halves: half the scores fall below it and half above it. It is written as \(M_d\).
For ungrouped data, the method depends on whether the number of scores is odd or even. When \(N\) is odd, the median is the value of the \(\left(\dfrac{N+1}{2}\right)^{\text{th}}\) score. When \(N\) is even, it is the average of the two middle scores.
For grouped data, the median is found with
\[M_d = L + \frac{\frac{N}{2} - F}{f_m} \times i\]
where \(L\) is the lower boundary of the class containing the median, \(F\) is the cumulative frequency before that class, \(f_m\) is the frequency of the median class, and \(i\) is the class width.
2.12.3 Mode
The mode is the score that occurs most often in a set of data. It is written as \(M_o\). If two scores tie for the highest frequency, the data are bimodal; if more than two do, the data are multimodal.
For ungrouped data, the mode is found simply by looking for the most repeated score.
2.12.4 When to Use Mean, Median or Mode
No single measure is best for every situation, and part of using statistics well is choosing the right one.
The mean uses every score in the data and feeds naturally into further statistical work, so it is preferred when the distribution is fairly symmetrical. Its weakness is that a few extreme scores can pull it far off-centre.
The median ignores the actual size of extreme scores and looks only at position, so it is the better choice when the data contain very high or very low scores, when the distribution is lopsided, or when the classes are open-ended (such as “70 and above”). It also suits qualitative data that can be ranked, such as levels of honesty or health.
The mode is the quickest to find and is the only measure that works for purely categorical data, such as the most common blood group in a class. It is useful when we simply want the most typical value.
2.13 Measures of Variability
A measure of central tendency tells us where the centre of a set of scores lies, but it hides something important: how spread out the scores are. Two groups can share exactly the same mean yet look completely different. In one, everyone scores close to the average; in the other, scores swing wildly from very low to very high.
To capture this spread we use a measure of variability, also called a measure of dispersion. It tells us how far the individual scores are scattered around the central value.
The syllabus deals with three such measures: the range, the variance and the standard deviation.
2.13.1 Range
The simplest measure of variability is the range, which we met while building a frequency distribution. As a measure of spread, it is the difference between the highest and the lowest score:
\[\text{Range} = \text{Highest score} - \text{Lowest score}\]
The range is quick and easy to find, which is its great strength. Its weakness is that it depends only on the two extreme scores and ignores everything in between, so a single unusually high or low score can make it misleading.
2.13.2 Variance and Standard Deviation
To use every score in measuring spread, we look at how far each score deviates from the mean. The standard deviation is the most dependable and widely used measure of variability, and the variance is its close companion.
The idea works in steps. First, find how far each score lies from the mean — its deviation, \(x = X - M\). Some deviations are positive and some negative, and they would cancel out if simply added. To prevent this, each deviation is squared, which removes the negative signs in a mathematically sound way. The average of these squared deviations is the variance. Taking the square root of the variance returns the measure to the original units of the scores, and this square root is the standard deviation, written with the Greek letter sigma, \(\sigma\).
For ungrouped data, the formulas are
\[\sigma^2 = \frac{\sum x^2}{N} \qquad \text{and} \qquad \sigma = \sqrt{\frac{\sum x^2}{N}}\]
where \(x = X - M\) is each score’s deviation from the mean, \(\sum x^2\) is the sum of the squared deviations, and \(N\) is the number of scores.
2.13.3 Importance of Central Tendency and Variability
Neither kind of measure is enough on its own; a full description of data needs both. Central tendency gives the representative score of a group, while variability shows how tightly or loosely the scores cluster around it.
When variability is low, the scores sit close to the centre, and the group is homogeneous — its members are alike on the trait measured. When variability is high, the scores are widely scattered, and the group is heterogeneous — its members differ greatly. Fig. 2.13-1 shows two groups with the same mean but very different spreads.
Diagram
Fig. 2.13-1 – two distribution curves centred on the same mean: one tall and narrow (low variability, homogeneous group), the other short and wide (high variability, heterogeneous group)
Return to Group A and Group B from Solved Example 2.7. Both had a mean of 60, but Group A was homogeneous and Group B heterogeneous. Reporting only the mean would have made the two groups look identical. It is the measure of variability that tells them apart.
2.14 Summary
Psychology studies behaviour scientifically — through methods that are empirical, objective and replicable. Introspection, the earliest method, declined because a person’s private self-reports could not be checked by anyone else.
Observation records behaviour by watching it closely. Naturalistic observation watches spontaneous behaviour in a real-life setting, giving high ecological validity, but it cannot establish cause and effect. Scientific observation, or field study, controls some factors so relationships can be examined more closely.
The longitudinal method follows the same people over a long period, tracing genuine change but at great cost and with drop-outs. The cross-sectional method compares different groups at a single time — quick and economical, but group differences may reflect generation rather than development.
The case study is an in-depth investigation of one individual, group or event, drawing on many sources. It offers great depth, but its findings cannot be generalised because a single case may not represent others.
The survey gathers information from many people through questionnaires, inventories or interviews. It has wide reach but only surface depth, and self-reports risk social desirability, where people give the expected answer rather than the true one.
The interview is a face-to-face verbal exchange that also lets the interviewer read non-verbal cues. It is direct and flexible, but depends heavily on the interviewer’s skill and can be shaped by their bias.
The experiment manipulates an independent variable, measures a dependent variable, and controls extraneous variables. It alone establishes a clear cause-and-effect relationship, though its controlled setting can be artificial and low in ecological validity.
Sampling studies a sample in order to draw conclusions about a whole population. Representative and random samples allow findings to be generalised; a biased sample does not.
A psychological test is a standardised tool for measuring behaviour. Good tests are reliable, valid and objective, and have proper norms. Tests are classified by the trait measured, nature, time and use of language, and are used for assessment, comparison, selection and research.
Ethical standards protect the people studied: voluntary participation, informed consent, debriefing, the sharing of results, and confidentiality of data.
Statistics is used to interpret research data. Descriptive statistics summarise the data at hand; inferential statistics generalise from a sample to a population and judge whether differences between groups are real or due to chance.
In psychology, statistics helps to describe and summarise data, compare individuals or groups, investigate relationships between variables, and predict.
Raw scores are organised by rank order, statistical tables, or frequency distributions. A frequency distribution groups scores into class intervals and shows how often scores fall into each.
A measure of central tendency gives a single representative value: the mean (the average, which uses every score but is pulled by extremes), the median (the middle value, best when extreme scores are present), and the mode (the most frequent value).
A measure of variability shows how spread out the scores are: the range (highest minus lowest), the variance (the average of the squared deviations from the mean), and the standard deviation (the square root of the variance). Both central tendency and variability are needed to describe data fully, because the same average can hide very different spreads.
2.15 Key Terms
- Aptitude test — a test that measures a person’s potential to succeed in a particular field or activity.
- Biased sample — a sample chosen for the researcher’s convenience or on purpose, which fails to represent the population.
- Case study method — an intensive, in-depth investigation of a single individual, group or event, using many sources; also called the case history method.
- Central tendency (measure of) — a single value that represents a whole set of scores by locating its centre.
- Class interval — a range of scores grouped together as one category in a frequency distribution.
- Class midpoint — the value at the centre of a class interval, found by averaging its two limits.
- Closed-ended question — a question offering fixed options to choose from.
- Confidentiality of data — the ethical duty to keep participants’ personal information private and secure.
- Confounding variable — an extraneous variable that is not controlled and so distorts an experiment’s results.
- Control group — the group in an experiment that does not receive the independent variable and acts as a baseline.
- Correlation — a measure of how strongly, and in what direction, two variables move together.
- Covert behaviour — inner mental activity, such as thoughts and feelings, that cannot be directly observed.
- Cross-sectional method — an observational method that compares different groups at a single point in time.
- Cumulative frequency — the running total of frequencies added through the classes of a distribution.
- Data — a set of numerical facts or scores collected for study.
- Debriefing — explaining a study’s true purpose to participants afterwards and ensuring they leave unharmed.
- Dependent variable — the behaviour measured in an experiment, whose value depends on the independent variable.
- Descriptive statistics — statistical methods that summarise and describe the data actually collected.
- Dispersion — another name for variability, the spread of scores around the centre.
- Ecological validity — the extent to which findings reflect genuine, real-life behaviour.
- Empirical — based on evidence gathered through observation or measurement rather than opinion.
- Ethical standards — the rules of responsible conduct that protect people taking part in research.
- Experimental group — the group in an experiment that receives the independent variable.
- Experimental method — a method that manipulates one variable and measures its effect on another while controlling all else, to establish cause and effect.
- Extraneous variable — any factor other than the independent variable that could affect the dependent variable and must be controlled.
- Field study — scientific observation carried out in a real-life setting with some factors controlled.
- Frequency — the number of times a score occurs in a set of data.
- Frequency distribution — a summary showing how often scores, or groups of scores, occur.
- Heterogeneous — describing a group whose members differ widely from one another.
- Homogeneous — describing a group whose members are similar to one another.
- Hypothesis — a specific, testable prediction about the outcome of a study.
- Independent variable — the factor the experimenter deliberately manipulates as the suspected cause.
- Inferential statistics — statistical methods that generalise from a sample to a population and judge whether results are due to chance.
- Informed consent — a participant’s genuine agreement to take part, given after being fully informed about the study.
- Intelligence test — a test used to measure a person’s intelligence.
- Interview — a face-to-face method in which an interviewer asks an interviewee spoken questions.
- Interview schedule — a fixed list of questions asked in the same order to every interviewee.
- Interviewee — the person who answers the questions in an interview.
- Interviewer — the person who asks the questions in an interview.
- Introspection — an early method of studying the mind by observing and reporting one’s own inner experiences.
- Inventory — a self-report tool of statements with fixed response choices, on which a person reports their own views or traits.
- Longitudinal method — an observational method that follows the same group of people repeatedly over a long period.
- Mean — the arithmetic average, found by dividing the sum of scores by their number.
- Median — the middle value of a set of scores arranged in order.
- Mode — the most frequently occurring score in a set of data.
- Naturalistic observation — the study of spontaneous behaviour in a natural setting, without interference.
- Non-verbal cues — wordless signals such as body language, eye contact and facial expression.
- Non-verbal test — a test that uses pictures rather than language.
- Norms — established reference values against which an individual’s test score is compared.
- Objectivity — freedom from the personal bias of the investigator.
- Observation — a method that records, perceives, analyses and interprets behaviour by watching it closely.
- Open-ended question — a question that respondents answer freely in their own words.
- Overt behaviour — behaviour that can be directly observed.
- Performance test — a test that requires the person to actually carry out a task.
- Personality test — a test used to identify a person’s personality traits.
- Population — the whole group about which a researcher wants to reach conclusions.
- Power test — a test with no time limit whose items rise from easy to difficult.
- Primary data — data collected first-hand by the researcher for the current study.
- Psychological test — a standardised, systematic tool for measuring a sample of behaviour.
- Questionnaire — a formalised, written set of questions used to collect information on a topic.
- Random sample — a sample in which every member of the population has an equal chance of being selected.
- Range — the difference between the highest and lowest scores.
- Raw data — scores in their crude, unorganised form; also called ungrouped data.
- Reliability — the quality of a test that gives consistent results when repeated.
- Replicability — the quality of a method that yields similar results when repeated by others.
- Representative sample — a sample that reflects all the important characteristics of the population.
- Sample — the smaller group, drawn from a population, that is actually studied.
- Sample survey — a survey carried out on a sample chosen to represent the population.
- Sampling — the process of selecting a sample from a population.
- Scientific method — a systematic, evidence-based procedure for studying behaviour that is objective and repeatable.
- Scientific observation — observation in which certain factors are controlled or manipulated to study their effect.
- Secondary data — data collected earlier by someone else and reused by the present researcher.
- Sharing results of the study — the ethical duty to make research findings available to the scientific community and to participants.
- Social desirability — the tendency of respondents to give the answer that looks good rather than the true one.
- Speed test — a strictly time-bound test.
- Standard deviation — the square root of the average of the squared deviations of scores from their mean.
- Standardisation — keeping the questions, administration and scoring of a test or survey uniform for everyone.
- Statistic — a single numerical value calculated from a sample, such as its average.
- Statistics — the branch of mathematics concerned with collecting, organising, analysing and interpreting numerical data.
- Survey method — a method that gathers information from many people by asking them a set of questions.
- Theory — a broad, general explanation of some behaviour, from which hypotheses are drawn.
- Validity — the quality of a test that truly measures what it claims to measure.
- Variability (measure of) — a value showing how spread out scores are around the central value.
- Variable — any characteristic or condition that can take different values and be manipulated, controlled or measured.
- Variance — the average of the squared deviations of scores from their mean; the square of the standard deviation.
- Verbal test — a test that requires the use of language to answer.
- Voluntary participation — the principle that people take part in research entirely of their own free will.
2.16 Points to Ponder
If the experiment is the only method that can firmly establish cause and effect, why do psychologists still rely on observation, case studies and surveys at all?
Naturalistic observation captures real behaviour but cannot explain it, while the experiment explains behaviour but in an artificial setting. Can a single study ever offer both realism and control at once?
Informed consent requires telling people what a study is about. But if participants know exactly what is being tested, might they behave differently — and does that make some truths impossible to study ethically?
The mean takes every score into account, yet a single extreme value can make it misleading. Does this mean the most complete measure is not always the most honest one?
Two groups can share the same average yet differ completely in their spread. If you could know only one — the central tendency or the variability of a group — which would tell you more about it?
A biased sample can lead to confident but wrong conclusions. In an age of online polls and viral surveys, how can you judge whether a widely shared “finding” rests on a sample worth trusting?
2.17 Objective Questions
The earliest method used primarily by the structural school of psychology to study behaviour was:
- Experiment
- Introspection
- Survey
- Case study [1 mark]
The method in which a researcher studies an entire population with respect to some psychological or social variable is the:
- Case study method
- Experimental method
- Survey method
- Longitudinal method [1 mark]
In an experiment, the variable that is deliberately manipulated by the experimenter is the:
- Dependent variable
- Extraneous variable
- Confounding variable
- Independent variable [1 mark]
A sample in which every member of the population has an equal chance of being selected is a:
- Biased sample
- Random sample
- Representative sample
- Convenience sample [1 mark]
Which of the following is NOT a characteristic of a good psychological test?
- Reliability
- Validity
- Objectivity
- Speed [1 mark]
Which measure of central tendency is most strongly affected by extreme scores?
- Mean
- Median
- Mode
- Range [1 mark]
Which of the following is NOT a role played by statistics in psychology?
- Describing and summarising data
- Comparing groups
- Manipulating data to fit the hypothesis
- Investigating relationships between variables [1 mark]
A self-report tool made up of statements with fixed response choices, on which a person reports their own views, is an:
- Interview schedule
- Inventory
- Observation record
- Experiment [1 mark]
The observational method that follows the same group of people repeatedly over a long period is the:
- Cross-sectional method
- Longitudinal method
- Field study
- Case study [1 mark]
Fill in the blank: The difference between the highest and the lowest score in a set of data is called the ______. [1 mark]
Fill in the blank: In an experiment, the behaviour that is measured, and whose value depends on the manipulated factor, is the ______ variable. [1 mark]
State whether True or False: A biased sample allows a researcher’s findings to be safely generalised to the whole population. [1 mark]
Match each measure in Column A with its meaning in Column B.
| Column A | Column B |
|---|---|
| a. Mean | i. the most frequently occurring score |
| b. Median | ii. the arithmetic average |
| c. Mode | iii. the spread of scores around the mean |
| d. Standard deviation | iv. the middle value of ordered scores |
[1 mark]
- Assertion (A): There was a decline in the use of introspection as a method. Reason (R): Introspection was a subjective method whose reports could not be checked by others.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true. [1 mark]
- Assertion (A): The experimental method can establish a cause-and-effect relationship. Reason (R): In an experiment, extraneous variables are controlled so that only the independent variable can affect the result.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true. [1 mark]
- Assertion (A): The median is often preferred over the mean when a set of scores contains a few extreme values. Reason (R): The median is not affected by the size of the extreme scores.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true. [1 mark]
2.18 Check Your Concepts
Define naturalistic observation. [2 marks]
What is a hypothesis? Give one example. [2 marks]
Name the three types of sample discussed in this chapter. [2 marks]
Define standard deviation. [2 marks]
Differentiate between primary data and secondary data. [3 marks]
Differentiate between a closed-ended and an open-ended question, giving one example of each. [3 marks]
Distinguish between the reliability and the validity of a psychological test. [3 marks]
Explain the difference between an independent variable, a dependent variable and an extraneous variable, using a single example. [4 marks]
Describe the main steps involved in conducting an experiment. [4 marks]
Differentiate between the longitudinal and the cross-sectional methods, and state one advantage of each. [4 marks]
Explain, with an example, why both central tendency and variability are needed to describe a set of scores fully. [4 marks]
Describe the experimental method in psychology. In your answer, explain the role of variables and control, and state one advantage and one limitation of the method. [6 marks]
Discuss the survey method. Explain its purpose, the tools it uses to collect data, and state one advantage and one limitation. [6 marks]
Explain the five ethical standards that must be followed in psychological research. [6 marks]
Compare descriptive and inferential statistics, and describe three ways in which statistics is used in psychology. [6 marks]
2.19 Case-Based Questions
- A psychologist wants to find out whether background noise affects how well students remember what they study. She arranges two groups of students. One group studies a passage in a quiet room; the other studies the same passage with mild traffic noise playing. Both groups are then tested on how much they recall.
- Which method of psychology is the researcher using?
- Identify the independent variable and the dependent variable in this study.
- Name one extraneous variable the researcher should control, and say why. [4 marks]
- A school counsellor is asked to help a Class X student who has recently become withdrawn and stopped completing schoolwork. The counsellor talks with the student, interviews the parents and class teacher, looks through the student’s earlier report cards, and gives a few standard tests, then puts all this together to understand what has changed.
- Which method of psychology is the counsellor mainly using?
- State one advantage and one limitation of this method in this situation. [4 marks]
- A television channel runs a poll on its website asking viewers whether they support a new city rule. Ninety per cent of those who respond say “yes,” and the channel announces that “the public supports the rule.”
- What kind of sample is this poll based on?
- Explain why the channel’s conclusion may be misleading.
- Suggest one way to make the sample more trustworthy. [5 marks]
- Two sections of Class XI wrote the same test. In Section A, the mean score was 60 and the standard deviation was 4. In Section B, the mean score was also 60, but the standard deviation was 16.
- What does the equal mean tell you about the two sections?
- What does the difference in standard deviation tell you about how the students in each section performed?
- Which section is more homogeneous, and why? [6 marks]
2.20 Practice With Data
For the scores 12, 15, 14, 15, 18, 15, 20, 11, 16, find the mean, the median and the mode. [4 marks]
For the scores 5, 7, 9, 11, 13, find the range and the standard deviation. [4 marks]
The frequency distribution below shows the scores of 20 students on a short test. Calculate the mean of the grouped data.
| Class Interval | Frequency (f) |
|---|---|
| 0–4 | 2 |
| 5–9 | 4 |
| 10–14 | 6 |
| 15–19 | 5 |
| 20–24 | 3 |
[4 marks]
Two groups took the same aptitude test. Group P had a mean of 50 and a standard deviation of 3. Group Q had a mean of 50 and a standard deviation of 12. State which group performed more consistently, and give a reason for your answer. [3 marks]
The frequency distribution below shows the marks of 26 students. Identify the median class and calculate the median.
| Class Interval | Frequency (f) |
|---|---|
| 10–19 | 3 |
| 20–29 | 6 |
| 30–39 | 10 |
| 40–49 | 5 |
| 50–59 | 2 |
[4 marks]