Summary Statistics: Averages and Spread

Summing up a whole data set

A list of numbers on its own is hard to make sense of, so statisticians describe a whole data set with just a couple of figures. The first figure is an average, a single value that stands for the typical member of the set. There are three averages to choose from: the mean, the median and the mode, and part of the skill is knowing which one to use. The second figure describes the spread, how stretched out or bunched up the values are, using the range or the interquartile range. This module shows how to find each average, how to choose fairly between them, what an outlier does to the mean, and how to measure spread, so that any data set can be summed up honestly.

Averages and spread words

Learn these five terms before you calculate and choose between them.

Match each average to when it is best

Mean, median or mode?

Each average suits a different kind of data. Pick the one that gives an honest picture.

Find the outlier

A data set is 3, 5, 6, 7 and 40. Which value is the outlier?

• 40
• 3
• 6
• There is no outlier

Two numbers tell the story

One figure is rarely enough to describe a data set. Two sets of marks can share the very same average and still be completely different: in one class the marks might be bunched tightly around the average, while in another they might be scattered from very low to very high. That is why a good summary gives an average AND a measure of spread together. The average says where the middle sits, and the spread says how reliable that middle is as a description of everyone. When you are asked to compare two data sets, always compare both: which one is higher on average, and which one is more spread out. Naming only one of the two throws away half the story.

Pick the true facts about averages

Select every statement about summary statistics that is true.

• The mean uses every value in the data
• The median is the middle value when the data is in order
• The mode is the value that appears most often
• The range is a type of average
• An outlier has no effect on the mean

Order how to find the median

Put these steps for finding the median in the correct order.

• Write out all the values
• Put the values in order of size
• Find the middle position
• Read off the value in the middle
• If two values share the middle, take their mean

Complete the averages facts

The _____ is the total divided by how many values there are. The _____ is the middle value once the data is in order. The value that appears most often is the _____. The highest value take away the lowest value gives the _____.

A classroom of pocket money

Picture a class where nearly everyone gets a small amount of pocket money each week, but one pupil happens to get a very large amount. If you add everyone up and share the total out evenly, that single large amount drags the shared figure upward, making it look as though the class is better off than most pupils really are. A fairer way to describe the typical pupil is to line everyone up from least to most and look at the pupil in the very middle, who is untouched by the one large amount at the end. This is exactly why statisticians keep more than one way of finding a typical value, and why they always ask which one gives the honest picture rather than the flattering one.

Tap the measures of spread

Tap every measure below that describes the spread of a data set, not its average.

• the range
• the interquartile range
• the mean
• the mode

Work out the range

A data set has a highest value of 20 and a lowest value of 3. The range is the highest value take away the lowest value. What is the range?

Choose the right measure

Read each situation and choose the best measure.

• A few houses on a street cost far more than all the rest. Which average best shows a typical price?
• A shop wants to know which shoe size to stock the most of. Which measure helps?
• Two classes have the same mean mark. What else should you compare?

Build a summary point

Choose the word for each gap to complete one accurate point about summarising data.

Explain averages and spread

A friend has a set of numbers and asks how to describe them with just a couple of figures. Explain clearly how to summarise a data set.

• Explain the three averages and how to find each one
• Explain when the median is fairer than the mean
• Explain what the range and the interquartile range measure
• Explain what an outlier is and its effect on the mean
• Explain why you should give both an average and a measure of spread