Mean, Median, Mode and Range

Four definitions, one sorting habit, and one idea about what an extreme value does. The arithmetic is trivial; the test is whether you compute the statistic that was asked for.

The four

Take the set 4, 8, 6, 4, 13.

Sorted: 4, 4, 6, 8, 13.

Mean is the total divided by how many. That is 35 divided by 5, which is 7.

Median is the middle value of the sorted list. With five values, the middle is the third, which is 6.

Mode is the value appearing most often, which is 4. A set can have two modes, or none at all if nothing repeats.

Range is the largest minus the smallest: 13 minus 4, which is 9.

Sort anyway. It makes three of the four immediate and it prevents the most common error in the topic, which is taking the middle of an unsorted list.

The even-numbered case

With an even count there is no single middle, so average the two middle values.

For 3, 7, 9, 12: the middle two are 7 and 9, so the median is 8.

Eight is not in the set, and that is fine. A median does not have to be one of the values. An answer choice that insists on a listed value is sometimes the distractor.

Consecutive integers

For consecutive integers, the mean and the median are the same: the middle number. Seven consecutive integers with a median of 10 run from 7 to 13. Five consecutive integers with a mean of 8 add up to 5 x 8 = 40. Consecutive odd integers step by 2: nine of them with a median of 17 run up to 17 + 4 x 2 = 25.

Quartiles

The interquartile range is Q3 minus Q1: the spread of the middle half of the data, as a box plot shows it. With Q1 = 20 and Q3 = 40, it is 20.

Mean against median

Enlarge

Zoom in

This is the part of the topic that carries the questions worth thinking about.

Take five salaries: 30, 32, 35, 38 and 200 thousand.

• Mean: 335 / 5 is 67.
• Median: 35.

The mean is nearly double every value except one. That single large figure - an outlier - pulled it there, because the mean divides a total in which the outlier is fully present. The median only asks which value sits in the middle, and the 200 occupies one position exactly like the 30 does.

The mean is sensitive to outliers. The median is resistant.

That is why questions ask which statistic "best represents" a set. If the data contains one extreme value, the answer is the median. If the values are evenly spread, the mean is fine and is the usual answer.

It also runs the other way, which is asked:

• Adding a value above the mean raises the mean. Below, lowers it.
• Adding a value at the mean leaves it unchanged.
• Adding values at the ends of a sorted list moves the median by at most one position, and often not at all.

Working backward

The same move as in the arithmetic reasoning lesson: mean times count is the total, and the total is what backward questions are built on.

Six values have a mean of 15. A seventh value is added and the mean becomes 16. What was the seventh value?

• Old total: 15 times 6 is 90.
• New total: 16 times 7 is 112.
• Seventh value: 22.

The intuition that the answer is 16, or 17, is wrong and both are choices. Lifting the mean by one point across seven values takes seven extra points, so the new value has to be 16 plus 6, which is 22.

Weighted means

When groups differ in size, totals up, counts up, divide once. Never average the averages of unequal groups. The averages lesson in Arithmetic Reasoning works this in full.

What you can skip

Across the 97 questions on this topic:

• Standard deviation, variance and percentiles never appear.
• Chart types. Histograms, box plots and stem-and-leaf plots come up about once each. Reading the numbers off them is the same four measures.
• Weighted means come up once in this topic; the section above is enough.

Where people lose points

Not sorting before taking the median.

Taking the middle position rather than averaging the middle two on an even-sized set.

Confusing mode and median, which is why the M-O-D-E and M-O-S-T mnemonic is worth the second it takes.

Assuming a set has exactly one mode. It may have two, or none.

Reporting the mean when the question asked which statistic is least affected by an extreme value.

Counting a repeated value once when computing the mean. Every occurrence counts.

Work one in under a minute

Find the mean, median, mode and range of 12, 7, 15, 7, 9, 10.

Sorted: 7, 7, 9, 10, 12, 15.

• Mean: the total is 60, and 60 divided by 6 is 10.
• Median: six values, so average the third and fourth - 9 and 10 - giving 9.5.
• Mode: 7, the only repeat.
• Range: 15 minus 7 is 8.

Sorting first made three of those immediate.

Where this leads

The mean-times-count identity is the same one behind every missing-value question in Arithmetic Reasoning, and probability uses the same habit of counting a whole set carefully.