Typical Statistics
Ever heard of median or mode? These are common statistics that are used to measure typical results in data analysis. We use these quite a bit when measuring results in life science, so let’s review the terms to refresh our memories.
First, lets begin by going over the median, mode, and mean. These three statistics are also known as the measures of center, as they give a value intended to be the middle of a set of data. Then we will have a look at percentiles.
Median
The median is the value in the data set that most nearly lies in the middle of the sample. It is the value that splits the ordered data into two equal halves.
Mode
The mode is the value in a sample that appears most frequently. Data sets do not always have a mode.
Mean
The mean is the most familiar measure of center. It is also called the average. The mean is the sum of the values divided by the number of values.
Continue reading below for explanations and examples.
Finding the Median
In order to find the median of a sample, we first need to arrange the values in order from smallest to largest. If there is an odd number of values in the sample, the middle value is the median. If there are an even number of values in the sample, the midway point between the two middle values is the median.
Example 1
Find the median of this sample:
5,4,13,6,21,18,2
First, arrange the values in increasing order:
2,4,5,6,13,18,21
We can see that 6 is the middle value in the sample, so 6 is the median.
Example 2
Find the median of this sample:
2,7,14,17,22,16
First, arrange the values in increasing order:
2,7,14,16,17,22
Since there are an even number of measurements, the median is between the two middle values. In this case, 14 and 16 are the two middle values, so the median is 15.
Mathematically, we can find this value by adding the two middle values
together and then finding half of the new value. For this example:
14+16 = 30
30/2 = 15
Mode
To find the mode of a sample, we have to find the value that shows up the most. If more than one value appears multiple times, the
sample could have more than one mode. If every value appears only once
in the sample, there is no mode.
Example 1
Find the mode of the sample:
2,11,16,4,13,17,11,20,7,4,10,16,4
To make it easier, let’s arrange the values in increasing order:
2,4,4,4,7,10,11,11,13,16,16,17,20
The number 4 appears three times, which is more than any other value, so 4 is the mode.
Example 2
Find the mode of the sample:
7,12,32,16,18,14,8,4,2
Because each number appears only once, this sample does not have a mode.
Mean
The mean is known as the “point of balance” of the data. The kind of mean we are using is known as the arithmetic mean. In general, it is a very natural measure of location. However, it is oversensitive to extreme values. When outliers are present, the mean can be a poor measure of central location. Nevertheless, the arithmetic mean is by far the most widely used measure of central location.
Example 1
Find the mean of the sample:
2,3,5,8,9,11,12,13,14,17
Add all the values and divide by the total number of values:
=(2+3+5+8+9+11+12+13+14+17) / 10
=94/10
=9.4
The mean of the sample is 9.4.
Example 2
Find the mean of the sample:
37,38,40,41,43,44,47,52,54
Add all the values and divide by the total number of values:
=(37+38+40+41+43+44+47+52+54) / 9
=396/9
=44
The mean of the sample is 44.
Percentiles
The pth percentile of a data set is the value such that p% of the observations are smaller than or equal to it. The median can be thought of as the 50th percentile.
Example
On our standardized test results, we see that we scored in the 88th percentile. What is the significance of this result?
Being in the 88th percentile means that 88% of the other students who took the test earned the same or a lower score than we did, and that 12% of the students scored higher than we did.
References
Doerge (2009). Statistics at the Bench- A Step-by-Step Handbook for Biologists by Doerge, Rebecca W – Bremer, Martina [Hardcover-spiral (2009)]. CSH, Hardcover(2009). ISBN 0879698578.
Rosner B (2010). Fundamentals of Biostatistics, 7 edition. Duxbury Press. ISBN 0538733497.
Samuels ML, Witmer JA and Schaffner A (2011). Statistics for the Life Sciences, 4 edition. Addison Wesley. ISBN 0321652800.