4.14 Comparison of Mean and Median

examples

Marci’s exam scores for her last math class were 79, 86, 82, and 94. What would the mean of these values would be?

The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

What is the mean number of TD passes?

Both examples are described further in the following video.

The following example illustrates one of the downfalls of the arithmetic mean.  The mean is not resistant to extremely high or low data values (outliers).

Example – Mean is not resistant to Outliers

One hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below.

Income (thousands of dollars) Frequency
15 6
20 8
25 11
30 17
35 19
40 20
45 12
50 7

We calculate the mean and find the mean household income of our sample to be 33.9 thousand dollars ($33,900).

Now, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand).  Adding the new family to our data.  Our new mean is now 83.1 thousand dollars ($83,069).

While 83.1 thousand dollars ($83,069) is the correct mean household income, it no longer represents a “typical” value.

Imagine the data values on a see-saw or balance scale. The mean is the value that keeps the data in balance, like in the picture below.

Drawing of a balance bar. A large blue block is on left end, and two smaller blue rectangles are on right end of balance point. One is close to the balance, one is further away.

If we graph our household data, the $5 million data value is so far out to the right that the mean has to adjust up to keep things in balance.

Drawing of a balance bar. A large blue block is on left end and two smaller blue rectangles are on also on left of balance point. On right, a small blue rectangle is significantly far away from the balance point.

For this reason, when working with data that have outliers – values far outside the primary grouping – it is common to use a different measure of center, the median.

Median

The median of a set of data is the value in the middle when the data is in order.

  • To find the median, begin by listing the data in order from smallest to largest, or largest to smallest.
  • If the number of data values, N, is odd, then the median is the middle data value. This value can be found by rounding N/2 up to the next whole number.
  • If the number of data values is even, there is no one middle value, so we find the mean of the two middle values (values N/2 and N/2 + 1)

example

Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

What is the median TD value?

Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7

Learn more about these median examples in this video.

Example

Let us return now to our original household income data

Income (thousands of dollars) Frequency
15 6
20 8
25 11
30 17
35 19
40 20
45 12
50 7

Here we have 100 data values. Since 100 is an even number, we need to find the mean of the middle two data values – the 50th and 51st data values. To find these, we start counting up from the bottom:

There are 6 data values of $15, so                  Values 1 to 6 are $15 thousand

The next 8 data values are $20, so                  Values 7 to (6+8)=14 are $20 thousand

The next 11 data values are $25, so                Values 15 to (14+11)=25 are $25 thousand

The next 17 data values are $30, so                Values 26 to (25+17)=42 are $30 thousand

The next 19 data values are $35, so                Values 43 to (42+19)=61 are $35 thousand

From this we can tell that values 50 and 51 will be $35 thousand, and the mean of these two values is $35 thousand. The median income in this neighborhood is $35 thousand.

[/hidden-answer]

If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is $35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is.

View more about the median of this neighborhood’s household incomes here.

 

In addition to the mean and the median, there is one other common measurement of the “typical” value of a data set: the mode.

Mode

The mode is the element of the data set that occurs most frequently.

The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed.

Example

In our vehicle color survey earlier in this section, we collected the data

Color Frequency
Blue 3
Green 5
Red 4
White 3
Black 2
Grey 3

Which color is the mode?

Mode in this example is explained by the video here.

It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once.

Try It

Reviewers were asked to rate a product on a scale of 1 to 5. Find

  1. The mean rating
  2. The median rating
  3. The mode rating
Rating Frequency
1 4
2 8
3 7
4 3
5 1

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