Application of Averages

In everyday life, the word "average" almost always refers to one of three values: the mean, the median, or the mode. Knowing how to calculate each one is only half the job. The other half, the part that shows up constantly in word problems, news reports, and real decision-making, is knowing which average to use and what it actually tells you about a set of data. That is what "application of averages" means: taking the tools you already have for mean, median, and mode and using them to answer a real question about a real data set.

Before applying these ideas, it helps to restate the three definitions clearly.

• Mean: add up every value and divide by how many values there are, \( \)mean\( = \frac{\)sum of values\(}{\)number of values\(} \). This is often just called the "average."
• Median: order the data from smallest to largest and find the middle value. If there is an even number of values, average the two middle numbers.
• Mode: the value (or values) that occur most frequently in the data set.

If you need a deeper refresher on how these three are calculated, see the lessons on mean and median and mode before working through the applications below.

Suppose seven students score the following on a quiz out of 100:

\( 78, \ 82, \ 82, \ 90, \ 95, \ 100, \ 55 \)

Mean: add the scores and divide by 7.

\( \frac{78+82+82+90+95+100+55}{7} = \frac{582}{7} \approx 83.1 \)

Median: order the scores, \( 55, 78, 82, 82, 90, 95, 100 \), and take the middle (fourth) value, which is \( 82 \).

Mode: the score \( 82 \) appears twice, more than any other score, so the mode is \( 82 \).

Here the mean, median, and mode are close together, so any of them gives a fair summary of how the class performed. This is a common situation when a data set does not contain extreme values.

Now compare that to a small company with six employees earning these yearly salaries, in thousands of dollars:

\( 35, \ 38, \ 40, \ 42, \ 45, \ 250 \)

The last value, \( 250 \), belongs to the owner and is far larger than the rest, this is an outlier.

Mean: \( \frac{35+38+40+42+45+250}{6} = \frac{450}{6} = 75 \) (thousand dollars).

Median: order the values, \( 35, 38, 40, 42, 45, 250 \), and average the two middle values, \( \frac{40+42}{2} = 41 \).

Mode: no value repeats, so this data set has no mode.

Notice the huge gap: the mean salary is \( 75{,}000 \), but five of the six employees earn far less than that. The single outlier has pulled the mean upward, so it does not represent a "typical" employee at all. The median, \( 41{,}000 \), is a much more honest description of what most workers actually earn. This is exactly why news reports about income or housing prices often quote the median rather than the mean. For more on how extreme values affect a data set, see the lesson on range and outliers.

The same reasoning used in these two examples shows up across many fields:

• Sports: a player's mean points per game summarizes overall performance, while the mode of results (win, loss, tie) shows the most common outcome.
• Weather: the mean daily temperature over a month gives a general climate picture, but a single unusually hot or cold day can shift it.
• Business: average sale price (mean) versus the median sale price can tell very different stories when a few very expensive items are included.
• Surveys and grades: the mode identifies the most common response or grade, which is useful when the data is categorical rather than numerical.

When a word problem asks you to describe or compare a data set, work through these questions:

• Does the data set contain any values that are much larger or smaller than the rest? If so, the mean may be misleading, and the median is usually a better summary.
• Is the data numerical or categorical? Mode works for both, but mean and median only make sense for numerical data.
• Does the problem ask for the "most common" value? That is a mode question, not a mean or median question.
• After calculating an average, always check whether it makes sense in context, an average number of siblings of \( 2.3 \) is reasonable to report even though no family literally has \( 2.3 \) children, because it still summarizes the group.

Practicing with mixed problems like these, and reviewing how mean, median, and mode relate to each other, builds the judgment needed to choose the right average every time. For a broader comparison of all three measures side by side, revisit the lesson on the center of a data set.

Related lessons

• Statistics Help