Statistics and Data Summary for the TMUA

Q: Why is the median often preferred over the mean for comparing salaries?

Salaries are often skewed by a few very high earners. These outliers pull the mean upwards, making it unrepresentative of the typical worker. The median remains at the center of the distribution and is unaffected by these extreme values.

Q: How do I calculate the midpoint for a class like $10 \leq x < 20$?

The midpoint is found by averaging the class boundaries: $\frac{10 + 20}{2} = 15$.

Q: Can the range be estimated from a cumulative frequency graph?

Yes. The estimated range is the difference between the $x$ value where the cumulative frequency is at its maximum and the $x$ value where the curve begins at zero frequency.

Updated August 2025

This guide covers the calculation and interpretation of measures of location and spread for both ungrouped and grouped data. You will learn to estimate statistics from frequency tables and cumulative frequency graphs, and how to use summary values like the median and interquartile range to make robust comparisons between different populations.

Core concept

Descriptive statistics summarise a population's characteristics using measures of location, such as the mean or median, and measures of spread, such as the range or interquartile range, allowing for like for like comparisons between data sets.

Summarising Ungrouped Data

For ungrouped data, statistics provide exact values for a data set. There are three primary measures of location and one primary measure of spread that you must master:

Mean: The arithmetic average, calculated by summing all values and dividing by the total number of values, $n$ . $\text{Mean} = \frac{\sum x}{n}$ .
Median: The middle value when data is arranged in ascending order. For $n$ values, the median is at the $\frac{n+1}{2}$ position.
Mode: The most frequently occurring value in the set. A data set can be bimodal if two values appear with the same highest frequency.
Range: The difference between the maximum and minimum values in the data set. $\text{Range} = x_{\text{max}} - x_{\text{min}}$ .

Worked Example

Find the mean, median, mode, and range for the data set: $3, 7, 7, 2, 11, 6$ .

First, order the data: $2, 3, 6, 7, 7, 11$ .

Mean: $\frac{2+3+6+7+7+11}{6} = \frac{36}{6} = 6$ .
Median: There are 6 values, so the median is between the 3rd and 4th values: $\frac{6+7}{2} = 6.5$ .
Mode: The value 7 appears twice, so the mode is 7.
Range: $11 - 2 = 9$ .

Grouped Data and Estimates

When data is presented in a frequency table with classes (e.g., $10 \leq x < 20$ ), we lose the original raw values. Consequently, any statistics calculated are estimates.

Modal Class: The class interval with the highest frequency density.
Estimated Mean: We assume all data points in a class take the value of the midpoint. $\text{Estimated Mean} = \frac{\sum f x}{\sum f}$ , where $f$ is the frequency and $x$ is the midpoint of the class interval.
Estimated Range: The difference between the upper boundary of the highest class and the lower boundary of the lowest class.

It is vital to understand why these are estimates: we do not know the exact distribution of values within each class. For instance, in a class $0 \leq x < 10$ with 10 values, all 10 could be 1, or all 10 could be 9. Using the midpoint 5 is a reasonable representative assumption, but rarely exact.

Graphical Representations

Cumulative frequency graphs are used to estimate the median and quartiles for grouped data.

Median ( $Q_2$ ): The value at the $50\%$ mark of the total frequency ( $n/2$ ).
Lower Quartile ( $Q_1$ ): The value at the $25\%$ mark ( $n/4$ ).
Upper Quartile ( $Q_3$ ): The value at the $75\%$ mark ( $3n/4$ ).
Interquartile Range (IQR): The difference between the upper and lower quartiles ( $Q_3 - Q_1$ ).

To find these values, draw a horizontal line from the required frequency on the $y$ axis to the curve, then drop a vertical line to the $x$ axis to read the estimated value.

Describing and Comparing Populations

Statistics allow us to describe a population's 'typical' value and its 'consistency'. To compare two distributions, you must always compare one measure of location and one measure of spread using like for like values.

Comparing Location: If Group A has a higher median than Group B, you can conclude that, on average, Group A has higher values.
Comparing Spread: If Group A has a smaller IQR than Group B, the values in Group A are more consistent and less spread out around the middle.

Advantages and Disadvantages of Summary Values

Mean: Uses every piece of data but is heavily distorted by outliers (extreme values).
Median: Not affected by outliers, making it a better measure of location for skewed data (e.g., house prices or salaries).
Mode: Only measure suitable for qualitative (non numerical) data, but there may be no mode or many modes.
Range: Simple to calculate but only uses two values and is highly sensitive to outliers.
IQR: Measures the spread of the middle $50\%$ of data. It is much more robust than the range because it ignores extreme outliers.

Key takeaways

Always order the data before finding the median for ungrouped sets.
Summary values for grouped data are always estimates because the raw data is unknown.
The Interquartile Range ( $Q_3 - Q_1$ ) is a more reliable measure of spread than the range because it is less affected by extreme outliers.
When comparing two data sets, you must compare a measure of location (e.g. median) and a measure of spread (e.g. IQR) to provide a full description.

Tips

In the TMUA, if a question involves a data set with a single extreme value, the mean will likely be a poor representative. Look for the median instead. Also, remember that a 'modal class' is an interval, not a single number.

Cautions

A common mistake is using the frequency ( $y$ axis) as the data value. The frequency tells you how many pieces of data are in a category, whereas the median and quartiles are always values from the $x$ axis.

Insight

Statisticians distinguish between 'resistance' and 'efficiency'. The mean is efficient because it uses all data points, but the median is resistant because it is not changed by moving extreme values further away from the center. Choosing between them depends on the quality and distribution of your data.

Worked Examples

Example 1

There are two sets of data: the mean of the first set is 15, and the mean of the
second set is 20.
One of the pieces of data from the first set is exchanged with one of the pieces of
data from the second set.
As a result, the mean of the first set of data increases from 15 to 16, and the mean of
the second set of data decreases from 20 to 17.
What is the mean of the set made by combining all the data?

16\frac{1}{4}

16\frac{1}{3}

16\frac{1}{2}

16\frac{2}{3}

16\frac{3}{4}

Practice Questions

Practice Question 1

In this question,

x_1, x_2, x_3, \ldots

is an arithmetic progression, all of whose terms are integers.
Let

n

be a positive integer. If the median of the first

n

terms of the sequence is an integer, which of the following three statements must be true?
I The median of the first

n + 2

terms is an integer.
II The median of the first

2n

terms is an integer.
III The median of

x_2, x_4, x_6, \ldots, x_{2n}

is an integer.

A:none of them

B:I only

C:II only

D:III only

E:I and II only

F:I and III only

G:II and III only

H:I, II and III

Frequently asked questions

Why is the median often preferred over the mean for comparing salaries?

Salaries are often skewed by a few very high earners. These outliers pull the mean upwards, making it unrepresentative of the typical worker. The median remains at the center of the distribution and is unaffected by these extreme values.

How do I calculate the midpoint for a class like $10 \leq x < 20$ ?

The midpoint is found by averaging the class boundaries: $\frac{10 + 20}{2} = 15$ .

Can the range be estimated from a cumulative frequency graph?