Mathematics II · Calculator allowed

Mean, Median, Mode, and Range

Four simple ways to describe a list of numbers, what's typical, what's in the middle, what shows up most, and how spread out it all is.

Imagine you wrote down how many hours you studied each day this week. That little list of numbers tells a story, but it's easier to talk about if you can boil it down to a single number or two. That's exactly what mean, median, mode, and range do.

These four words sound fancy, but each one is just a small, friendly calculation. This is the calculator-allowed part of the CAEC math test, so feel free to punch the numbers in, the important thing is knowing which calculation to do. Let's meet all four.

The four measures, in plain English

  • Mean, the everyday "average." Add up all the numbers, then divide by how many numbers there are.
  • Median, the middle value once you put the numbers in order from smallest to largest.
  • Mode, the number that appears most often. A list can have one mode, more than one, or none at all.
  • Range, how spread out the numbers are: the largest value minus the smallest value.
Memory trick: median is the one in the middle, and mode is the one that shows up the most. Mean is the "mean old average" that makes you do the most work.

One data set, four different answers

Let's use a single example list of test scores and find all four measures. Here it is, already sorted from smallest to largest:

478891012Median = 8(the middle value)8 appears twice → Mode = 8

Seven scores, lined up in order. The middle one is the median, and the repeated value is the mode.

Worked example #1: all four for one list

Our data set is the seven test scores from above: 4, 7, 8, 8, 9, 10, 12. Already in order, that always helps. Let's find each measure.

  • Mean: add them up: 4 + 7 + 8 + 8 + 9 + 10 + 12 = 58. There are 7 numbers, so 58 ÷ 7 = 8.28... ≈ 8.3.
  • Median: with 7 numbers, the middle one is the 4th value. Counting in: 4, 7, 8, 8, 9, 10, 12. The median is 8.
  • Mode: the only value that repeats is 8 (it appears twice). The mode is 8.
  • Range: largest minus smallest: 12 − 4 = 8.
Data:   4, 7, 8, 8, 9, 10, 12   (7 numbers, in order)

Mean   = (4+7+8+8+9+10+12) ÷ 7
       = 58 ÷ 7
       = 8.28...  ≈ 8.3

Median = the 4th value (the middle)
       = 8

Mode   = the value that repeats
       = 8

Range  = 12 − 4
       = 8
Mean ≈ 8.3, Median = 8, Mode = 8, Range = 8. Notice the mean does not have to be one of the numbers in the list, and that's perfectly normal.

Worked example #2: finding the median of an even list

When there is an even number of values, there are two middle numbers instead of one. The median is the mean of those two, you average the pair in the middle.

Take the data set 3, 6, 7, 10, 11, 14. There are 6 values, so the two in the middle are the 3rd and 4th: 7 and 10.

  • Find the two middles: 3, 6, 7, 10, 11, 14. The middle pair is 7 and 10.
  • Average them: (7 + 10) ÷ 2 = 17 ÷ 2 = 8.5.
Data:   3, 6, 7, 10, 11, 14   (6 numbers, in order)

Middle pair = 7 and 10
Median = (7 + 10) ÷ 2
       = 17 ÷ 2
       = 8.5
Median = 8.5. Just like the mean, the median of an even list can land between two whole numbers. That is expected, not a mistake.

Worked example #3: how an outlier pulls the mean

An outlier is a value that sits far away from the rest of the data. Outliers tug hard on the mean but barely move the median. Watch what happens to the same little list of weekly earnings when one big number sneaks in.

Without the outlier
Data: 20, 22, 25, 27, 26

Mean   = 120 ÷ 5 = 24
Median = 25  (middle of
         sorted list)

Mean and median both sit right in the middle of the pack, around 24 to 25.

With one outlier (200)
Data: 20, 22, 25, 27, 26, 200

Mean   = 320 ÷ 6 ≈ 53.3
Median = 25.5  (avg of
         25 and 26)

The single big value yanks the mean up past 53, but the median barely budges.

See the difference? Adding one unusually large value (200) more than doubled the mean, from 24 to about 53.3, even though almost everyone earned in the twenties. The median moved only from 25 to 25.5. That is why the median is often the fairer "typical" value when a data set has outliers, think house prices or incomes, where a few huge numbers can distort the average.

Tips that keep you out of trouble

  • Always sort first for median and mode. Put the numbers in order from smallest to largest before you hunt for the middle or the most common value. The mean and range don't care about order, but the median definitely does.
  • Count how many numbers you have. An odd count has one clear middle value; an even count means you average the two middle values for the median.
  • Don't forget repeats when finding the mean. If a number appears twice, add it twice, and count it twice when you divide.
  • Watch for "no mode" and "more than one mode." If nothing repeats, there is no mode. If two different values tie for most frequent, the list has two modes.

Your turn: practice problems

Grab a calculator and work each one. Remember to sort the list before looking for the median or mode. No peeking until you have tried.

  1. Find the mean, median, mode, and range of: 5, 8, 8, 11, 13
  2. Find the median of this even list: 2, 4, 9, 10
  3. Find the range of: 14, 3, 22, 9, 7
  4. A class scores 6, 7, 7, 8, 90 on a quiz. Find the mean and the median, then say which one better describes a typical score.
Tap to reveal the answers
  • 1. List (already sorted): 5, 8, 8, 11, 13. Mean = (5 + 8 + 8 + 11 + 13) ÷ 5 = 45 ÷ 5 = 9. Median = middle value = 8. Mode = 8 (it repeats). Range = 13 − 5 = 8.
  • 2. Sorted: 2, 4, 9, 10. Even count, so average the middle pair (4 and 9): (4 + 9) ÷ 2 = 13 ÷ 2 = 6.5.
  • 3. Largest is 22, smallest is 3. Range = 22 − 3 = 19. (You don't need to fully sort for the range, just spot the biggest and smallest.)
  • 4. Sorted: 6, 7, 7, 8, 90. Mean = (6 + 7 + 7 + 8 + 90) ÷ 5 = 118 ÷ 5 = 23.6. Median = middle value = 7. The median (7) describes a typical score better, because the outlier 90 pulls the mean way up to 23.6, far above almost everyone.

Why this matters for the CAEC

Mean, median, mode, and range come up all the time on the calculator-allowed part of the CAEC math test, often inside real world questions about scores, prices, temperatures, or survey results. Knowing which measure to use, and spotting when an outlier is hiding, lets you answer quickly and confidently.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.

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