How to Find the Mean, Median, Mode, Range, and Standard Deviation

By Jessica Smith
Students can use surveys to collect data for analysis.

Statistics is the science of collecting and analyzing numerical data. You could take a survey or collect information and then use measures like mean, median, mode and range to see what that data mean. Standard deviation is often the next calculation for describing what a set of numerical data is doing. Once you know how to find mean, you can find standard deviation.



Mean Is the Average

Mean is a measure of central tendency. It measures what the majority of the data are doing toward the middle of a set. The mean is often referred to as the average of a data set. As an example, an algebra class has 10 students. Their grades on the last test were 85, 90, 87, 93, 100, 53, 78, 85, 99 and 82. What is the average grade for the students? To find mean, simply add all the numbers in a data set and divide by the number of items in the set:

85 + 90 + 87 + 93 + 100 + 53 + 78 + 85 + 99 + 82 = 852 852 / 10 = 85.2

The average, or mean, test grade in the class is 85.2.

Mode Occurs Most

Mode is another measure of central tendency. The mode is just the number that occurs most frequently. It's easy to remember because mode and most sound alike. Using the algebra class example, what grade occurred most frequently among the students? To answer, put the values in order:

53, 78, 82, 85, 85, 87, 90, 93, 99, 100

The only grade that occurred more than once is 85. Since 85 occurred most, the mode is 85.

Median Is the Middle, Range Is the Spread

Median is another measure of central tendency. The median is simply the middle number of a set. Put the numbers in order and look for one in the middle. If there is no middle number, add the two in the center and divide by 2. In the algebra class example, what is the median grade? To answer, put the values in order:

53, 78, 82, 85, 85, 87, 90, 93, 99, 100

Since there are an even number of test grades, there is no middle number. The two test grades in the middle are 85 and 87. Add them and divide by 2:

85 + 87 = 172 172 / 2 = 86

The median, or middle grade, is 86.

Range is a quick calculation. Range is simply the largest value minus the smallest. It shows you how spread out the numbers are. For these grades, subtract 53 from 100 to get the range of 47.

Find Variance Before Standard Deviation

Standard deviation is the square root of the variance, so you must find the variance first. Variance is the average of the squared difference of each number from the mean. That may sound confusing, but it's pretty simple to do. Take each number in the set and subtract if from the mean. Then square it. Add those values together, and divide by the number of items in your set. Working with the algebra class grades again, subtract each one from the mean:

85.2 - 53 = 32.2 85.2 - 78 = 7.2 85.2 - 82 = 3.2 85.2 - 85 = 0.2 85.2 - 85 = 0.2 85.2 - 87 = -1.8 85.2 - 90 = -4.8 85.2 - 93 = -7.8 85.2 - 99 = -13.8 85.2 - 100 = 14.8

Square each of those values, then add them together:

1,036.84 + 51.84 + 10.24 + 0.04 + 0.04 + 3.24 + 23.04 + 60.84 + 190.44 + 219.04 = 1,595.6

Finally, divide that sum by the number of items in the set, in this case 10:

1,595.6 / 10 = 159.56

The variance for this data set is 159.56.

Standard Deviation Measures Spread

Standard deviation is the measure of how spread out the numbers are from the center of a data set. A small standard deviation means a lot of the numbers are grouped around the middle of the set. A large standard deviation means that the number are spread out with some very high and low numbers. With the algebra grades, use this equation:

square root (159.56) = 12.63

The standard deviation for this data set is 12.63.

About the Author

With hands-on experience in the traditional classroom, the online setting, and the world of curriculum development, Jessica Smith is a veteran educator who is passionate about learning. Smith earned a M.Ed. in curriculum and instruction from Concordia University and is certified in mathematics and exceptional student education.