Mean, Median or Mode? Choosing the Right Average

A company says, "The average salary here is high." The workers say, "Not for us." Both can be telling the truth, because there is more than one average. This is a brief note on Averages.

Three averages, three questions

• Mean: the fair share. Add all the values and divide by how many there are.
• Median: the middle value, when the values are sorted. With two in the middle, take the halfway value.
• Mode: the value that appears most often. It also works for words.

Two tricks with the mean

• Total = mean $\times$ count. It finds a missing value: the total you need, take away the total you have.
• Two groups together. Add the two totals and divide by everyone. Halfway between the two means is right only when the groups are the same size.

When to use which

If every value goes up by 10, all three averages go up by 10. If every value doubles, all three double.

A worked example

The salaries of nine people in a small firm, in thousands of rupees, are $18, 19, 21, 22, 22, 24, 25, 27$ and $160$.

1. Mean. The total is $338$, so the mean is $\dfrac{338}{9} = 37.6$ thousand rupees.
2. Median. The values are sorted, and the 5th is $22$ thousand rupees.
3. Mode. $22$ appears twice, so the mode is $22$ thousand rupees.
4. Which to report. Eight of the nine earn between 18 and 27 thousand rupees. The one high salary drags the mean above all of them, so the median is the honest typical salary.

Practice with new numbers

A program to try

The program finds the three averages of the worked example.

Mistakes to avoid

• Using the mean when one value is far from the others. The extreme value drags it.
• Using the mean for categories. You cannot add colours.
• Forgetting to sort before finding the median. The middle of an unsorted list means nothing.
• Giving the mode as a count. The mode is the value itself, not how many times it appears.

Check yourself

1. The marks of nine students are $35, 38, 40, 41, 42, 44, 45, 46$ and $98$. Which average best describes how the class did?

2. Can the mean be a number that is not in the data? Can the median? Can the mode?

3. A student says that the mean of $2, 4$ and $9$ is $4$, because $4$ is in the middle. What is wrong?

4. A team of 6 has a mean height of $150\ \text{cm}$. A new player of $171\ \text{cm}$ joins. What is the new mean height?