Let's Practice:

Find the Mean, Median and Mode of the following numbers:

1)   15, 18, 21

2)   16, 18, 20, 22

3)   5, 6, 7, 8, 9

4)   2, 4, 6, 8, 10, 12
5)   100, 200, 300, 400

6)   20, 30, 40, 50, 60

7)   23, 35, 32

8)   136, 140, 141

9)   94, 94, 98, 98 

10) 6, 8, 7

Mode:

The most frequently occurring number or numbers in a set of numbers.

Range: 

The difference between the greatest and the least in a set of numbers. 

Median:

In a set of numbers arranged in order from least to greatest, the middle number, or the average of the two middle numbers.

Mean:

The average
Mean, median, and mode are three of the most common kinds of "averages". 

The "mean" is the "average".
How do you find the mean or the average?
You add up all the numbers and then divide by the number of numbers. 

The "median" is the "middle" value in the list of numbers.
To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. Find the number in the very middle.

The "mode" is the value that occurs most often. 
If no number is repeated, then there is no mode for the list.

The "range" is just the difference between the largest and smallest values.

  • Find the mean, median, mode, and range for the following list of values:
      • 13, 18, 13, 14, 13, 16, 14, 21, 13


      The mean is the usual average, so:
      (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

    Note that the mean isn't a value from the original list. This is a common result. 
    You should not assume that your mean will be one of your original numbers.


    The median is the middle value, so I'll have to rewrite the list in order:

      13, 13, 13, 13, 14, 14, 16, 18, 21

    There are nine numbers in the list,
    so the middle one will be the 
    (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:

      13, 13, 13, 13, 14, 14, 16, 18, 21

    So the median is 14.

    The mode is the number that is repeated more often than any other.
    Which number is repeated most often?

    13 is the mode.

    The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

      mean: 15
      median:       14
      mode:       13
      range: 8

Note: The formula for the place to find the median is 
( [the number of data points] + 1) ÷ 2

If that seems confusing, then you can just count in from both ends of the list until you meet in the middle. Remember to do whatever is easier for your. Either way will work.

  • Find the mean, median, mode, and range for the following list of values:
      • 1, 2, 4, 7

    The mean is the usual average:

    (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5

    The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers. In this case, the median is the mean (the usual average) of the middle two values:

      (2 + 4) ÷ 2 = 6 ÷ 2 = 3

    The mode is the number that is repeated most often, but all the numbers in this list appear only once, so there is no mode.


    The largest value in the list is 7, the smallest is 1, and their difference is 6, so the range is 6.

      mean: 3.5
      median:       3
      mode: none
      range:       6

The list values were whole numbers, but the mean was a decimal value. Getting a decimal value for the mean (or for the median, if you have an even number of data points) is perfectly okay; don't round your answers to try to match the format of the other numbers.

  • Find the mean, median, mode, and range for the following list of values:
      •  8, 9, 10, 10, 10, 11, 11, 11, 12, 13

    The mean is the usual average:

      (8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5

    The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5th value; that is, I'll need to average the fifth and sixth numbers to find the median:

      (10 + 11) ÷ 2 = 21 ÷ 2 = 10.5

    The mode is the number repeated most often. This list has two values that are repeated three times.

    The largest value is 13 and the smallest is 8, 
    so the range is 13 – 8 = 5.

      mean: 10.5
      median:       10.5
      modes:       10 and 11
      range: 5

While unusual, it can happen that two of the averages (the mean and the median, in this case) will have the same value.


About the only hard part of finding the mean, median, and mode is keeping straight which "average" is which. Just remember the following:

    mean: regular meaning of "average"
    median: middle value
    mode: most often

  • A student has gotten the following grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall class grade. What is the minimum grade he must get on the last test in order to achieve that average?

    • The unknown score is "x". Then the desired average is:

      (87 + 95 + 76 + 88 + x) ÷ 5 = 85

    Multiplying through by 5 and simplifying, I get:

      87 + 95 + 76 + 88 + x = 425 
                            346 + x = 425 
                                      x = 79

      He needs to get at least a 79 on the last test.