Using the Standard Deviation
You made a number of observations about the data sets for the school activities. You used mean and median to measure the center of the data, and you used the interquartile range (IQR) to measure the spread.
When outliers are present, the median and IQR are used to measure center and spread because they are unaffected by extreme values. When the data appears to be symmetric and there are no known outliers, the mean and standard deviation (another measure of spread) are used.
In the past, you have used the mean absolute deviation.
Deviation can be defined as “how far from the normal.” The mean absolute deviation (MAD) shows the distance of the average data point from the mean.
The standard deviation (σ) is very similar to the MAD. It also is a measure of spread that uses the mean of the data. Standard deviation is used much more often than MAD in daily statistics because it allows comparisons to the overall data set and is very useful for comparing percentiles. It also maintains the unit of measure of the original data. Here is the formula for standard deviation:
σ = Greek letter sigma that signifies standard deviation in statistics Σ = take the sum of
x = each data value
x̄ = mean of all the data values
n = number of data values
5 steps to calculate the standard deviation:
1. Subtract the mean from each data point.
2. Square the differences found in step 1.
3. Add the squares from step 2.
4. Divide the sum from step 3 by the number of data values.
5. Find the square root of the number from step 4.
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