The mean or average of a data set is one representation of the center of the data set. It is one measure of center. The others are the median and the mode. To calculate the mean, add all the data points together, then divide by the number of data points.
mean =number of valuessum of values
Suppose a data set represents the heights of different towers. The mean of this data set gives an idea of a typical height. Calculating the mean could be seen as rearragning the blocks so that all the towers have the same height.
After the blocks are rearranged, the towers each have a height of 4. Therefore, the mean is 4. If the heights are written as x, then the mean is sometimes written as xˉ. The towers' mean height can then be written as xˉ=4.
When on vacation in Mexico, Peter finds a rose species he has never seen before. He decides to study how many petals each flower has. The result of his study is this. 10,14,11,9,16 How many petals do the flowers on average have?
A measure of spread is a way of quantifying how spread out, or different, the points in a data set are. A small spread means data points are similar, while a large spread means they are different. This is illustrated by the two data sets below. Both have a mean, median and mode of 3, but, we can assume the second data set has a larger spread because of how different its data points are.
Standard deviation is a commonly used measure of spread. It is a measure of how much a randomly selected value from a data set is expected to differ from the mean. To denote the standard deviation, the Greek letter σ is used, which is read as "sigma."To calculate a standard deviation, the rule σ=n(x1−xˉ)2+(x2−xˉ)2+…+(xn−xˉ)2 is used, where n is the number of values in the data set and xˉ is the mean of the set.
The standard deviation, σ, of a data set is calculated using the rule σ=n(x1−xˉ)2+(x2−xˉ)2+…+(xn−xˉ)2, where n is the number of values in the data set and xˉ is the mean of the set. Performing this calculation in one step makes for a convoluted expression. Therefore, it is best divided into a few, smaller steps. Consider the following data set as an example. 1,5,3,8,3,12
For each data value, x−xˉ can now be calculated and added to a table. This shows how much each data point varies from the mean.
x | x−xˉ |
---|---|
1 | 1−4=-3 |
5 | 5−4=1 |
3 | 3−4=-1 |
8 | 8−4=4 |
3 | 3−4=-1 |
12 | 12−4=8 |
Square the deviations, and add them to a new column in the table.
x | x−xˉ | (x−xˉ)2 |
---|---|---|
1 | -3 | (-3)2=9 |
5 | 1 | 12=1 |
3 | -1 | (-1)2=1 |
8 | 4 | 42=16 |
3 | -1 | (-1)2=1 |
12 | 8 | 82=64 |
Finally, take the square root of the just found quotient to get the standard deviation. Here, the fraction is used instead of the quotient, to avoid rounding errors. σ=692≈3.92 Thus, a randomly chosen value from this data set is expected to deviate roughly 4 units from the mean.