Data Handling and Descriptive Statistics

Every number throws something away

A psychologist finishes a study with a page of individual results and reports a single number. That number is a summary, and every summary is a decision about what to keep and what to lose. The mean uses every value but tells you nothing about the shape. The median keeps the middle and ignores how extreme the extremes were. The range keeps the spread and loses everything in between. None of that is a fault; it is what summarising is. The skill this topic tests is knowing what each one keeps, so that you can choose the summary that answers the question and say what it left behind.

Kinds of data

Six terms. The first four come in two pairs, and questions often turn on telling one pair from the other.

Summary to what it keeps

Quantitative or qualitative?

A researcher records how many words each participant recalled, and also writes down what each said about how they tried to remember. Which describes the two sets of data?

• The word counts are quantitative and the comments are qualitative
• The word counts are qualitative and the comments are quantitative
• Both are quantitative, because both were recorded systematically
• Both are qualitative, because both describe what participants did

Where the data came from

The other pair is about origin rather than form, and it matters because it changes what you can and cannot claim.

Work out the mean

Five participants score 5, 12, 13, 14 and 16 on a memory test. Work out the mean of their scores.

Work out the median

Using the same five scores of 5, 12, 13, 14 and 16, put them in order and work out the median.

Work out the range

Using the same five scores of 5, 12, 13, 14 and 16, work out the range.

The gap between mean and median

You just calculated a mean of 12 and a median of 13 for the same five people, and that gap is worth a moment. Four of the five scored 12 or above. The single score of 5 pulled the mean down below all four of them, because the mean uses every value and a low one drags it. The median did not move, because it only cares which value sits in the middle. Neither number is wrong. They are answers to different questions, and reporting only the mean here would give a slightly gloomier picture of the group than four of the five deserved.

Plot the fifth participant

A scatter diagram shows hours slept the night before against score on a memory test. Four participants are already plotted. **Plot the fifth participant**, who slept 7 hours and scored 8.

What the scatter shows

On that diagram the points rise from lower left to upper right. What does the display tell you, and what does it not?

• That the two variables are associated, but not that one caused the other
• That sleeping longer causes a higher score, since the pattern is clear
• Nothing, because a scatter diagram is only a way of drawing data
• That the mean score is higher for those who slept longer than for those who did not

Displays and what they suit

Select the TWO statements that correctly match a display to what it is for.

• A bar chart is used to compare totals across separate categories
• A scatter diagram is used to show the relationship between two variables
• A frequency table is used to show the cause of a difference between groups
• A histogram is used to display data that falls into separate unrelated categories

Numbers in a paragraph

Data in the form of numbers is _____, while data in the form of words is qualitative. Data you collected yourself is _____ data, and data collected by someone else is secondary. To summarise a set of scores you might use the mean, which uses every value, or the _____, which is unaffected by a single extreme result. To describe how spread out the scores were you would use the _____. And when you report a figure, giving more significant figures than your data supports claims a _____ you do not have.

A results section, marked

Question: five participants scored 5, 12, 13, 14 and 16 on a memory test. Summarise the results. Model answer: the mean score was 12 and the median was 13, out of a possible 20. The range was 11, which is wide for only five participants and reflects one score of 5 that sits well below the rest. Because the mean uses every value, that single low score pulls it below the median, so the median is arguably the better summary of a typical participant here. The data are quantitative and primary, since they are scores collected in this study. With only five participants any summary should be treated cautiously, and quoting the mean to several decimal places would suggest a precision the data does not support. Notice what earns credit. Two summaries rather than one, a reason for preferring one of them, the data described in the specification own terms, and a note on what the numbers cannot support.

Claiming precision you do not have

Three of these statements report data honestly. Select the ONE that claims a precision the data cannot support.

• The mean score was 12 out of 20, based on five participants.
• The mean score was 12.0000 out of 20, based on five participants.
• The range was 11, which is wide given how few participants took part.
• The median was 13, and was not affected by the single low score.

Report your results

A psychologist tests eight people on how many words they recall after a short delay, and also asks each one how they tried to remember. Explain how you would summarise and present these results.

• State which of the two sets of data is quantitative and which is qualitative
• State whether the data is primary or secondary, and say how you know
• Name the summary you would use for a typical score, and say what it keeps and what it loses
• Name one measure of spread you would report, and say what it adds
• Name a display you would use, and say what it would show that a number alone would not
• Explain one thing you would be careful about when reporting the figures