In order to measure variability we will move from the sum of squares to the variance and then to the standard deviation.
Sum of squares is a good measure for overall variability,
but the size of the sum of squares is dependent upon the number of scores.
As you increase sample size the sum of squares will also increase,
but the increase is not due to increased variability simply from having more scores.
Therefore, we should standardize this measure and calculate the average
variability by dividing the sum of squares by the number of scores, or N.
This average sum of squares value is called the variance.
Variance is sometimes abbreviated as var,
but more commonly as SD^2 for reasons that will become apparent in a moment when we talk about the standard deviation.
Variance is a legitimate measure of variability,
but it has limitations.
First of all, variance is hard to interpret.
Variance is based on squared deviations, so it is an unrealistically large number.
Also, variance is based on squared units that are hard to think about. If we were talking about age
the variance might be 4 squared years.
What does that mean? What is a squared year?
So how can we get from squared units to non-squared units?
The solution is the standard deviation.
When a number is squared taking the square root of a number returns it to the original units.
Now we can talk about variability in years instead of squared years.
Since the variance is squared, we can unsquare it using the square root.
This unsquared variance is the standard deviation,
abbreviated SD.
The standard deviation is the square root of the variance.
We can interpret our data using the standard deviation.
Scores differ by an average of a standard deviation if the mean is 6 and the standard deviation is 2,
then one standard deviation above the mean is 6 + 2, or 8.
Two standard deviations above the mean would be 6 + 2 + 2, or 10.
One standard deviation below the mean would be 6 - 2, or 4.
Plus the standard deviation is in the same scale as the mean, not squared units like variance.
The standard deviation tells us how consistent and close the scores are.
A small standard deviation means that the scores are close and consistent.
A large standard deviation means that the scores are far apart and inconsistent.
The standard deviation also effects the shape of the distribution.
Larger standard deviations are more platykurtic; the scores are more spread out.
Smaller standard deviations are more leptokurtic; the scores are closer together.
The standard deviation gives us additional information that is not provided by the mean.
Without this additional information provided by the standard deviation
the mean could create a misleading conclusion.
For example, the average high temperature in San Francisco is
63.9 degrees, very similar to the average temperature in Kansas City, Missouri of 65.9.
The average low is 50.7 in San Francisco and
47.4 in Kansas City.
So San Francisco and Kansas City must have very similar climates, right?
Not at all. I have also included Springfield, Missouri in this table about three hours south of Kansas City, Missouri.
The average high and average low temperatures are still similar to San Francisco and to Kansas City, but
Springfield, Missouri has the most weather variety in the nation based on temperature variation, precipitation, wind,
and severe weather.
Number two in weather variety is Topeka, Kansas about two hours west of Kansas City.
San Francisco, California, on the other hand, is the top city with the least weather variety.
The temperature and climate is very consistent in San Francisco.
Like Goldilocks eating Baby Bear's porridge it is not too hot and not too cold here
in Springfield, Missouri, however,
we can have university classes canceled for a snowstorm on Monday,
shirt sleeve weather in the low 70s on Friday, and
Thunderstorms this weekend.
When it comes to weather, we're pretty rangy here at Springfield, Missouri.
For the people in San Francisco the weather is pretty much the same from day to day.
We can use standard deviation for making inferences about a sample.
Suppose that you are the coach for a football team.
You have two running backs, both of whom average five yards per carry.
Running back A has a standard deviation of 2. Running back B has a standard deviation of 10.
You were in a situation of third and goal with 1 yard to go, which running back would you go with?
Running back A.
He consistently gets the short yardage even on his worst carry 2 standard deviations below
his mean, he's still likely to get that 1 yard.
But what about a situation of third and goal with 10 yards to go?
You should go with running back B.
Yes, he may lose yardage, but he is much more likely to get the 10 yards that you need.
So let's quickly review the sum of squares, variance, and standard deviation all
represent the same thing: the variability in the data.
First you compute the sum of squares,
then you divide it by N to get the variance. And then you take the square root of the variance to get the standard deviation.