Mean and Median

Brokk Toggerson; Aidan Philbin

Mean and Median

Note to reader

We assume that you are probably familiar with the concepts of median and mean. The key point is that we will use the symbol x for the sample mean and μ for the population mean.

If you are unfamiliar, the section below from OpensStax Introductory Business Statistics by A. Holmes, B. Illowsky, and S. Dean is provided for your reference.

The “center” of a data set is also a way of describing location. The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. Technically this is the arithmetic mean. We will discuss the geometric mean later. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts meaning an equal number of observations on each side. The weight of 25 people are below this weight and 25 people are heavier than this weight. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean” and “average” is technically a center location. Formally, the arithmetic mean is called the first moment of the distribution by mathematicians. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”

When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “x bar”): $\stackrel{-}{x}$ .

The Greek letter μ (pronounced “mew”) represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

To see that both ways of calculating the mean are the same, consider the sample:
1; 1; 1; 2; 2; 3; 4; 4; 4; 4; 4

$\stackrel{-}{x}=\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7$

$\stackrel{-}{x}=\frac{3\left(1\right)+2\left(2\right)+1\left(3\right)+5\left(4\right)}{11}=2.7$

In the second calculation, the frequencies are 3, 2, 1, and 5.

You can quickly find the location of the median by using the expression $\frac{n+1}{2}$ .

The letter n is the total number of data values in the sample. If n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then $\frac{n+1}{2}$ = $\frac{97+1}{2}$ = 49. The median is the 49^th value in the ordered data. If the total number of data values is 100, then $\frac{n+1}{2}$ = $\frac{100+1}{2}$ = 50.5. The median occurs midway between the 50^th and 51^st values. The location of the median and the value of the median are not the same. The upper case letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median.

Calculating mean and median

AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;
Calculate the mean and the median.

The calculation for the mean is:

$\stackrel{-}{x}=\frac{\left[3+4+\left(8\right)\left(2\right)+10+11+12+13+14+\left(15\right)\left(2\right)+\left(16\right)\left(2\right)+\text{...}+35+37+40+\left(44\right)\left(2\right)+47\right]}{40}=\mathrm{23.6}$
To find the median, M, first use the formula for the location. The location is:
$\frac{n+1}{2}=\frac{40+1}{2}=20.5$
Starting at the smallest value, the median is located between the 20^th and 21^st values (the two 24s):
3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;

$M=\frac{24+24}{2}=24$

Which is better median or mean?

Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the “center”: the mean or the median?

$\stackrel{-}{x}=\frac{5,000,000+49\left(30,000\right)}{50}=129,400$

M = 30,000

(There are 49 people who earn $30,000 and one person who earns $5,000,000.)

The median is a better measure of the “center” than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.

Type your examples here.

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Physics 132 Lab Manual by Brokk Toggerson and Aidan Philbin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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