Math by First Principles
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Mean, Median, Mode, and Range

Learning goals

  • Compute the mean, median, mode and range of a list
  • Explain why the mean is the balance point where deviations cancel
  • Sort before taking a median, and average the two middle values
  • Read the range as spread, not as a center
  • Choose median over mean when an outlier drags the data
  • Name the mode as the only summary that works for non-numeric data

The mean: a fair share

The mean, also called the arithmetic average, answers a simple question. If you pooled all the values together and split the total evenly, how much would each one get? You find it by adding every value and dividing by how many values there are.

mean=sum of the valuesnumber of values.\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}.

Say five friends are carrying 44, 88, 66, 22, and 55 dollars. Together they hold 4+8+6+2+5=254 + 8 + 6 + 2 + 5 = 25 dollars. If they pour it all into one pile and redistribute it equally, each friend walks away with

255=5 dollars.\frac{25}{5} = 5 \text{ dollars}.

So the mean is 55 dollars. Notice what “fair share” really means here: replacing every value by the mean leaves the total untouched, because five fives still add to 2525. That is the defining property of the mean, and it is just the formula read backwards. That reading works because the sum equals the number of values times the mean.

Why the mean sits at the balance point

Calling the mean a “center” deserves a reason, not just a name. Here is the precise sense in which the mean balances the data. The deviation of a value is how far it sits above or below the mean. Values larger than the mean have a positive deviation, and values smaller than the mean have a negative one. The claim is that these deviations always cancel out exactly.

The deviations from the mean always add to zero#

Call the values x1,x2,…,xnx_1, x_2, \ldots, x_n, so there are nn of them, and let mm stand for their mean. By the definition of the mean, the sum of the values divided by nn equals mm. Multiplying both sides by nn gives a fact we will use in a moment: the values add up to n×mn \times m.

Now form the deviation of each value, xi−mx_i - m, and add all the deviations together:

(x1−m)+(x2−m)+⋯+(xn−m).(x_1 - m) + (x_2 - m) + \cdots + (x_n - m).

Separate the two kinds of term. All the values collect into their sum x1+x2+⋯+xnx_1 + x_2 + \cdots + x_n. The mean mm is then subtracted once for each of the nn values, so those subtractions remove n×mn \times m in total:

(x1+x2+⋯+xn)−n×m.(x_1 + x_2 + \cdots + x_n) - n \times m.

But the values add up to exactly n×mn \times m, so this becomes n×m−n×m=0n \times m - n \times m = 0. The deviations cancel completely, for any data set at all.

Picture the values as weights sitting on a ruler. The mean is then the point where you would put your finger to hold the ruler level. Every value below the mean tips the ruler one way and every value above it tips the ruler the other way. Because the deviations sum to zero, the two sides exactly balance. That is what makes the mean a true center of the data. The mean is the one point where the total pull from the smaller values matches the total pull from the larger ones.

The mean as the balance point of 2, 3, 4, 7A ruler from 0 to 8 with weights at 2, 3, 4, and 7 and a fulcrum at the mean 4; the deviations plus 3 and minus 3 cancel.-2-10+3234701235678mean = 4
The data 2, 3, 4, 7 balanced on a ruler. The fulcrum sits at the mean, 4. The value at 7 is 3 units above the mean, which exactly balances the values at 2 and 3, together 3 units below it, so the deviations cancel.

Worked example 1 Find the mean of 66, 99, 99, 44, and 77

Add the five values:

6+9+9+4+7=35.6 + 9 + 9 + 4 + 7 = 35.

There are 55 values, so divide the sum by 55:

mean=355=7.\text{mean} = \frac{35}{5} = 7.

The mean is 77. As a check, the two nines sit 22 above 77 for +4+4 in total. The 66 and the 44 sit 11 and 33 below the mean for −4-4 in total, and the 77 is right on the mean. The deviations cancel, exactly as the balance argument promised.

Check your understanding

Over four days a shop sold 1212, 1515, 99, and 88 sandwiches. What was the mean number sold per day?

The median: the middle value

The median is the value that lands in the middle once the data is sorted from least to greatest. It splits the ordered list into two equal halves. At least half the values sit at or below the median, and at least half sit at or above it. To find it, always sort first, then locate the middle.

When there is an odd number of values, one value sits in the dead center, and that value is the median. Sort 3,8,5,9,43, 8, 5, 9, 4 into 3,4,5,8,93, 4, 5, 8, 9; with five values the third one is the middle, so the median is 55. Two values fall below it and two fall above.

When there is an even number of values, no single value is in the center, and two values share the middle. The median is then the mean of those two middle values. Sort 7,2,10,47, 2, 10, 4 into 2,4,7,102, 4, 7, 10; the two middle values are 44 and 77, so the median is

4+72=5.5.\frac{4 + 7}{2} = 5.5.

A quick way to find the middle position: with nn values in sorted order, the median sits at position n+12\frac{n + 1}{2}, counting from either end. For n=5n = 5 that is position 33, a single value. For n=4n = 4 that is position 2.52.5, which falls between the 22nd and 33rd values, so you average that pair. Sorting is not optional, because “middle” means middle in order, not the middle of the list as it happened to be handed to you.

Finding the median for an odd and an even countTop row of five sorted values with the single middle value highlighted; bottom row of four sorted values with the two middle values highlighted and averaged.Odd count: one middle value34589median = 5Even count: average the middle two24710median = 5.5
The median splits the sorted data in half. With five values (top) the single middle value, 5, is the median. With four values (bottom) the two middle values 4 and 7 are averaged, giving 5.5.

Worked example 2 Find the median of 1414, 33, 99, 1414, and 66

Sort the five values from least to greatest:

3,  6,  9,  14,  14.3, \; 6, \; 9, \; 14, \; 14.

There are 55 values, an odd count, so the median is the single middle value at position 5+12=3\frac{5 + 1}{2} = 3. Counting in to the third value gives

median=9.\text{median} = 9.

Two values (33 and 66) lie below it and two (1414 and 1414) lie above, so 99 splits the data evenly. Notice that the repeated 1414s caused no trouble. The repeats did not matter because the median only cares about position, not about how large the largest values happen to be.

Check your understanding

Find the median of 88, 22, 1111, and 55.

The mode: the most common value

The mode is the value that appears most often. Where the mean and median do arithmetic to find a center, the mode just counts. Tally how many times each value occurs, and the value with the highest tally is the mode.

In the list 4,7,4,2,9,44, 7, 4, 2, 9, 4 the value 44 appears three times while every other value appears once, so the mode is 44. A data set can have more than one mode if several values tie for the most appearances. The list 1,1,5,8,81, 1, 5, 8, 8 has two modes, 11 and 88, because each of those two values appears twice. And a data set can have no mode at all when every value appears the same number of times, as in 3,6,93, 6, 9, where nothing repeats.

The mode is the only one of these summaries that also works for data that is not numbers. Suppose you record the favorite color of everyone in a class. You cannot add those colors together or sort them down the middle, but you can still ask which color comes up most. That winner is the mode.

The range: how spread out the data is

The first three numbers all try to pin down the center of the data. The range answers a different question: how spread out is it? The range is the distance from the smallest value to the largest, found by subtracting the minimum from the maximum.

range=maximum−minimum.\text{range} = \text{maximum} - \text{minimum}.

For the test scores 72,88,95,60,8172, 88, 95, 60, 81, the largest is 9595 and the smallest is 6060, so the range is 95−60=3595 - 60 = 35. A small range means the values are bunched closely together; a large range means they are stretched far apart. Two classes can share the very same mean and still feel completely different. One class where everyone scores near 8080 has a small range. A second class with scores running from 4040 to 100100 has a large range even if its mean is also 8080. The range is a first, rough measure of that spread.

Worked example 3 Find the mean, median, mode, and range of 5,8,3,8,65, 8, 3, 8, 6

Start with the mean. Add the five values and divide by 55:

mean=5+8+3+8+65=305=6.\text{mean} = \frac{5 + 8 + 3 + 8 + 6}{5} = \frac{30}{5} = 6.

For the median, sort the data into 3,5,6,8,83, 5, 6, 8, 8. With five values the middle one is the third, so

median=6.\text{median} = 6.

For the mode, count repeats. The value 88 appears twice and everything else once, so the mode is 88.

For the range, subtract the smallest value from the largest:

range=8−3=5.\text{range} = 8 - 3 = 5.

So this data set has mean 66, median 66, mode 88, and range 55. The mean and median agree here because the data is fairly even; the next section shows what happens when it is not.

Check your understanding

For the data 10,4,7,4,10,410, 4, 7, 4, 10, 4, which statement is true?

Mean versus median: which center to trust

When the data is lopsided, the mean and the median can tell very different stories, and knowing why lets you pick the honest one. The mean uses the actual size of every value, so a single far-off value, an outlier, drags the mean toward that extreme. The median uses only position, so one outlier barely moves it.

Picture five households with yearly incomes, in thousands of dollars, of 3030, 3535, 4040, 4545, and 10001000. The mean income is

30+35+40+45+10005=11505=230,\frac{30 + 35 + 40 + 45 + 1000}{5} = \frac{1150}{5} = 230,

which is larger than four of the five incomes. Reporting “the average income is 230230 thousand” would mislead anyone, because nobody here lives like that except the single wealthy household. The median tells a fairer story. The data is already sorted, so the middle value is

median=40,\text{median} = 40,

which sits right among the typical households. The lone outlier of 10001000 pulled the mean up by almost 200200 but moved the median not at all.

This is the practical rule of thumb. When the data is roughly even, the mean and median land close together and either one describes the center well. When the data is skewed by a few extreme values, the median is usually the more trustworthy center. The mean is still the right tool when you genuinely care about the total being shared out. Splitting a bill or finding a true per-person amount is one of those cases.

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Progressively harder sets of questions. Each opens on its own page.

Level 1 Foundational 12 questions Start → Level 2 Intermediate 12 questions Start → Level 3 Challenge 12 questions Start →

Free Response Questions (FRQ)

Longer questions in parts, to be worked out on paper. Progressive hints, the answer on its own so you can check yourself and try again, then the full worked solution, plus a rubric to mark your own work against.

Free response Work it out on paper 5 questions Start →
More practice (optional)

Extra sets, as hard as the Challenge set. Each one opens on its own page.

More resources (optional)

Other explanations of this lesson, if you want a second take.

A bit of history (Optional)

A cargo ship is caught in a storm. It is riding far too low. To save it, the crew heaves part of the cargo over the side. The ship limps into port. Whose loss is the cargo now at the bottom of the sea?

The old answer is that everyone pays. Traders around the Mediterranean, the sea south of Europe, refused to leave one unlucky owner carrying the whole loss. The cost was added up and then shared out among every merchant whose goods came home safe. Roman law had that rule fifteen hundred years ago, and shipping still uses it under the name general average.

Our word comes straight from that rule. Sailors called the shared cost the avaria, and average is what English made of it. So the word did not begin as a summary of a list. It began as one total, split fairly, so that each part bore an equal share.

That is the mean, long before anyone called it a mean. Add the values, split the sum evenly, and hand every value the same share. The total does not budge, just as five fives still add to twenty-five.