Data Analysis and Descriptive Statistics

Making sense of the numbers

A study might collect hundreds of results. Descriptive statistics are the tools that summarise that data so you can see what it shows at a glance. Two jobs matter most: finding a typical value (an average - the mean, median or mode) and seeing how spread out the data is (the range). Get those right, pick the correct one for your data, and display it in the right graph - that is what Paper 2 rewards.

Four measures to know

Three are averages (measures of central tendency); one measures spread (dispersion). Learn them, then the next steps put them to work.

Match each measure to its meaning

Find the mode

Seven participants recalled this many words: 4, 6, 6, 7, 6, 9, 5. What is the mode?

• 6, because it appears more often than any other value
• 7, because it is in the middle
• 5, because it is the smallest common value
• 9, because it is the highest

Calculate the mean

Five students scored 7, 9, 8, 6 and 10 on a memory test. Work out the mean score (add them up, then divide by 5).

Which average, and when?

They are not interchangeable - each has a best use:\n\n- Mean: uses every value, so it is powerful - but a single outlier (an extreme value) can drag it off and make it misleading. - Median: ignores how extreme the end values are, so it is the safer choice when the data is skewed or has outliers. - Mode: the only average that works for categories (like favourite colour) or for finding the most common result.

The outlier problem

A set of reaction times is mostly around 200 ms, but one participant scored 900 ms. Which average best represents the typical result?

• The median, because it is not distorted by the one extreme value
• The mean, because it uses every value
• The range, because it shows the highest and lowest
• No average can be used with an outlier

Two kinds of data

Before you analyse, know what kind of data you have - it decides which tools you can use.

Match each display to its use

Which is quantitative?

Select the TWO examples that are QUANTITATIVE data (numbers you can count or measure).

• The number of words each participant recalled
• Each participant's reaction time in milliseconds
• A written description of how a participant felt
• An interview transcript about someone's experience

How to find the median

Put the steps for finding the median of a data set in the right order.

• Collect all the values in the data set
• Put the values in order, from smallest to largest
• Find the value in the middle of the ordered list
• If there are two middle values, take the mean of those two
• State that middle value as the median

Choose the right tool

Three data situations. Pick the best statistic or display for each.

• Most incomes in a sample are modest, but one participant is a millionaire. Which average best shows the typical income?
• You want to show whether hours revised are linked to test scores. Which display is best?
• Your data is the favourite colour chosen by each participant. Which average can you even use?

Which average for the recall test?

A researcher records how many words each of ten participants recalled in a memory test. Explain which measure of central tendency they should report and why, and name one suitable way to display the data.

• Name the measure you would use (mean, median or mode) for this number data
• Justify your choice - for example the mean uses all the data, but the median is safer if there is an outlier
• Say what you would check the data for before deciding (an outlier or a skew)
• Name a suitable graph or table to display the results, and why it fits the data

Choose the average, and say why

A weaker answer just calculates an average. A grade-9 answer chooses the right one and says why. Remember the trade-off: the mean uses every value but is thrown off by an outlier; the median rides out outliers and skew; the mode is your only option for categories. And match the display to the data - a scatter diagram for a relationship, a histogram for continuous data, a bar chart for separate categories. Justifying the choice is the mark-winner.