Even More About Standard Deviation

 Like the other four measures of dispersion, the standard deviation gets smaller as the scores get more homogeneous, and larger the more heterogeneous they become. A small standard deviation indicates the scores are quite similar to the mean (closer to the peak); a large standard deviation says the score vary from the mean.

 

This measure of dispersion is calculated by taking the square-root of variance. Regardless of whether you used N or N-1 to calculate variance, standard deviation is the square-root of variance. If variance is 7.22, the standard deviation is 2.69. If variance is 8.67, the standard deviation equals 2.94.

Technically, the square-root of a population variance is called sigma and the square-root of a sample variance is called the standard deviation. As a general rule, population measures use Greek symbols and sample parameters use English letters. Since we tend to use large samples, we’ll focus on the standard deviation.

STANDARD

The standard deviation is “standard” in the sense that it takes steps of equal distance from the mean. Think of it as standing at the mean and taking 3 steps in one direction. If doesn’t matter if you step toward the high end or the low end. It only takes three steps to get from the mean to the end of a distribution. If you start at the mean and go toward the positive end, you’re there in 3 steps; and it’s 3 steps from the mean to the lowest end of the distribution. So the entire distribution is comprised of 6 steps (3 positive steps and 3 negative steps).

STEPS (Deviations From The Mean)

 

Each of these steps is the equal in distance but accounts for a different amount of people. The normal curve is like a mountain. If you’re standing on top of the mountain, your first step is always your largest. In a frequency distribution of a normally distributed variable, your first step accounts for the most people. Because most scores are close to the mean, most scores fall within plus or minus one standard deviation from the mean.

In fact, that’s our definition of normal. Normal is being close to the mean. Normal musical ability is scoring at the mean plus or minus one standard deviation. Normal basketball throwing is at the mean, plus or minus one standard deviation.

In a normally distributed variable, the percentages are consistent, regardless of what is being measured. Starting from the mean, the first step accounts for just over 34% of the scores. The next steps has 14% and the last step has 2%. Since normal frequency distributions are symmetrical, the percentages work on either side of the man. So the entire distribution looks like this: 

 

 

 

NOW YOU CHOOSE:
    Day 3: Dispersion
    A Bit More About Dispersion
    Even More About Dispersion
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    How To Calculate
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    Formulas For Dispersion
    Practice Problems
    More Practice Problems
    Basic Facts About Dispersion
    Vocabulary
    Quiz 3
    Summary