Comparing the three averages

What you will be able to do

• find the mean, the median and the mode of one table, and say what each tells you;
• use $3 \times \text{median} = \text{mode} + 2 \times \text{mean}$, and say when it fails;
• say which averages can be found when the last class is open.

Three answers for one table

Sixty shoppers were timed in a shop. The classes are $10$ minutes wide.

• Mean. The class marks are $5, 15, \dots, 65$ and $\sum fx = 1730$, so $\bar{x} = \dfrac{1730}{60} = 28.83$ minutes.
• Median. The cumulative frequencies are $3, 15, 35, 49, 56, 59, 60$. Half of $60$ is $30$, so the median class is $20$ to $30$, and $\text{median} = 20 + \dfrac{30 - 15}{20} \times 10 = 27.5$ minutes.
• Mode. The modal class is $20$ to $30$, and $\text{mode} = 20 + \dfrac{20 - 12}{40 - 12 - 14} \times 10 = 25.71$ minutes.

The three numbers are different, and each is true. The mode says what the most common visit lasts. The median says that half of the shoppers stay less than $27.5$ minutes. The mean is pulled up a little by the few shoppers who stay a long time.

The bars rise quickly and fall slowly. Data like this are skewed to the right. The long tail pulls the mean most, the mode least, and the median goes between: mean above median above mode. For data skewed to the left, the order is reversed.

The rule of thumb

For data that are moderately skewed, the three averages are linked by

This is an empirical relation: it comes from looking at many data sets, not from a proof. It says that the median is one third of the way from the mean to the mode. Try it on the shop data:

The true mode is $25.71$, so the rule is off by less than $1$ minute. It is useful to find a missing average. But it is only an approximation.

When the rule fails. Take a table with two peaks. Thirty values fall in the classes $0$ to $10$, $10$ to $20$, $20$ to $30$, $30$ to $40$ and $40$ to $50$ with frequencies $10, 3, 2, 3, 12$. The mean is $26.33$ and the median is $30$. The rule gives a mode of $3 \times 30 - 2 \times 26.33 = 37.33$. The mode by the formula is $44.29$. The rule is $7$ units away, because the data are far from a smooth single peak.

Which average should I use?

House prices and incomes are usually reported by the median, because a few very large values would lift the mean. A shop that stocks shoes uses the mode. A teacher who adds marks uses the mean.

An open class

A table may end with a class such as "$60$ and above". It has no upper end, so it has no class mark. Then the mean cannot be found. The median can be found if the median class is not the open class, because it only needs the frequencies. The mode can be found when the open class is not the modal class, because the formula only needs the frequency of a neighbouring class.

Practise with new numbers

Try it in Python

This program finds all three averages of the shop table and tests the rule of thumb. At the end it asks the statistics module for the median and the mode of a list of class marks.

Common mistakes

• Treating the rule as exact. It is an estimate, close only for moderately skewed data with one peak.
• Swapping the factors. The median has the factor $3$ and the mean has the factor $2$.
• Finding the mean when the last class is open. The mean needs a class mark for every class.
• Choosing the mean for data with extreme values. Report the median, and say why.

Remember these

• Mean, median and mode answer different questions about what is typical.
• Data with one peak and a long tail on the right usually have mean above median above mode.
• $3 \times \text{median} = \text{mode} + 2 \times \text{mean}$ is a rule of thumb, not a law.
• With an open last class the mean cannot be found.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A data set has mean $40$ and median $38$. Use the rule of thumb to estimate the mode.

2. In question 1, is the data skewed to the right or to the left?

3. A reporter says that the median house price in a town is lower than the mean. What does this tell you about the prices?

4. A table ends with the class "$80$ and above". Can you find the mean? Can you find the median?

5. Why is the rule $3 \times \text{median} = \text{mode} + 2 \times \text{mean}$ only an approximation?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the relation between the three averages.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics: skewness and the mean, the median and the mode.