What measure of variability is the square root of the average of the squared deviations from the mean?

Variance and standard deviation are measures of variability. The most common way we measure variability is by using the standard deviation.  However, when working with standard deviations we must first make sure that our data are normally distributed (otherwise we need to modify the way we look at distributions).  We most often think of distributions being "normal" when they look like the classical "bell curve" of quantitative data.  However it is very important for you to understand the idea of normality will apply to categorical data as well, although normality will be assessed in slightly different ways.  

But let's first take a quick look at standard deviations and variability.  We will take a look at how we calculate these by hand.  Subsequently we will discuss the assessment of normality for both quantitative and categorical data. 

The standard deviation is actually the square root of the variance. But wait a minute.. what is the variance? 

Don't get discouraged, we will walk through these calculations for you, and Susie, computing these values by hand. After this lesson, you will always be computing standard deviation using software such as Minitab Express.

Let's start step by step: 

  1. Step 1: Compute the sample mean (which we have already done):
  2. \(\overline{x} = \dfrac{\sum x}{n}\)

  3. Step 2:Calculate the deviance (or how far away each individual observation is away from the mean) by subtracting the sample mean from each individual value:
  4. \(x-\overline{x}\), these are the deviations

  5. Step 3:For each deviation, multiply it by itself (in other words, you are squaring the deviation):
  6. \((x-\overline{x})^{2}\), these are the squared deviations

  7. NOTE: This is an important step.  If you are curious you can skip step 3 and jump right to step 4.  What you will find is if you add up all of the deviation scores from step 2, the sum will always be equal to zero!  If you like algebra you can play with the formula for the mean and the total deviation scores and see that they are equivalent!

  8. Step 4: Find the sum of squares by summing of all of the squared deviations from Step 3:
  9. \(\sum (x-\overline{x})^{2}\), this is the sum of squares

  10. Step 5: Divide the sum of squares by \(n-1\) (where n is the total number of observations):
  11. \(\dfrac{\sum (x-\overline{x})^{2}}{n-1}\), this is the sample variance \((s^{2})\)

  12. Viola!  You have calculated the variance.  That was a lot of math.  Fortunately, you never have to do this by hand (but it is very interesting to see that variance is a simple matter of subtracting, multiplying, adding, and dividing!
  13. But this still hasn't got us to the standard deviation.  So we need to add one last step.  
  14. Step 6: Take the square root of the sample variance:
  15. \(\sqrt{\frac{\sum (x-\overline{x})^{2}}{n-1}}\), this is the sample standard deviation

  16. Viola!  The standard deviation! Just as the definition indicates, this is simply the square root of the variance
  17. NOTE: The reason this is the square root is because of Step 3 above.  Because of the math involved we had to square the deviation scores.  This results in the magnitude of the variance always being much larger than the original units of measurements in our data.  By taking the square root of the variance, the variability of the data can be expressed in the original measurement units by using the standard deviation.

So to keep track of some of the vocabulary introduced:

DeviationAn individual score minus the mean.

Sum of Squared DeviationsDeviations squared and added together. This is also known as the sum of squares or SS.

Sum of Squares\(SS={\sum \left(x-\overline{x}\right)^{2}}\)

VarianceApproximately the average of all of the squared deviations; for a sample represented as \(s^{2}\)

Standard DeviationRoughly the average difference between individual data values and the mean. The standard deviation of a sample is denoted as \(s\). The standard deviation of a population is denoted as \(\sigma\).

Sample Standard Deviation\(s=\sqrt{\dfrac{\sum (x-\overline{x})^{2}}{n-1}}\)

Statistical Indices of Data Variability

What measure of variability is the square root of the average of the squared deviations from the mean?

Measures of Dispersion

Range
The range gives you the most basic information about the spread of scores. It is calculated by the difference between the lowest and highest scores.

Interquartile Range:
The difference between the score representing the 75th percentile and the score representing the 25th percentile is the interquartile range. This value gives you the range of the middle 50% of the values in the data set.

Variance and Standard Deviation:
The standard deviation is the square root of the average squared deviation from the mean. The average squared deviation from the mean is also known as the variance.

Understanding and Calculating the Standard Deviation
Computers are used extensively for calculating the standard deviation and other statistics. However, calculating the standard deviation by hand once or twice can be helpful in developing an understanding of its meaning.

Calculating the variance and standard deviation
Consider the observations 8,25,7,5,8,3,10,12,9.

  1. First, determine n, which is the number of data values.
  2. Second, calculate the arithmetic mean, which is the sum of scores divided by n. For this example, the mean = (8+25+7+5+8+3+10+12+9) / 9 or 9.67
  3. Then, subtract the mean from each individual score to find the individual deviations.
  4. Then, square the individual deviations.
  5. Then, find the sum of the squares of the deviations...can you see why we squared them before adding the values?
  6. Divide the sum of the squares of the deviations by n-1. This is the Variance!
  7. Take the square root of the variance to obtain the standard deviation, which has the same units as the original data.

Sum of squared dev =  320.01

*Deviation = Score - Mean

Score Mean Deviation* Squared
Deviation
8 9.67 -1.67 2.79
25 9.67 +15.33 235.01
7 9.67 -2.67 7.13
5 9.67 -4.67 21.81
8 9.67 -1.67 2.79
3 9.67 -6.67 44.49
10 9.67 +.33 .11
12 9.67 +2.33 5.43
9 9.67 -.67 .45

  Standard Deviation = Square root(sum of squared deviations / (N-1)
    = Square root(320.01/(9-1))
    = Square root(40)
    = 6.32

Raw score method for calculating standard deviation
Again, consider the observations 8,25,7,5,8,3,10,12,9.

  1. First, square each of the scores.
  2. Determine N, which is the number of scores.
  3. Compute the sum of X and the sum of X-squared.
  4. Then, calculate the standard deviation as illustrated below.
Score   X2  
8 64  
25 625  
7 49 N=9
5 25  
8 64 Sum of X=87
3 9  
10 100  Sum of X2=1161
12 144  
9 81  
--- ---  
87 1161  
Standard Deviation = square root[(sum of X2)-((sum of X)*(sum of X)/N)/(N-1)]
    = square root[(1161)-(87*87)/9)/(9-1)]
    = square root[(1161-(7569/9)/8)]
    = square root[(1161-841)/8]
    = square root[320/8]
    = square root[40]
    = 6.32

Even simple statistics, such as the standard deviation, are tedious to calculate "by hand".


Copyright © 1997 T. Lee Willoughby

Which measure of variability is the average of the squared deviations from the mean?

The variance is the average of squared deviations from the mean. A deviation from the mean is how far a score lies from the mean. Variance is the square of the standard deviation. This means that the units of variance are much larger than those of a typical value of a data set.

What measure of variability is the square root of the average of the squared deviations from the mean group of answer choices?

chapter 2.

Is the square root of the average of the squared deviations from the mean?

The standard deviation is the square root of the average squared deviation from the mean. The average squared deviation from the mean is also known as the variance. Computers are used extensively for calculating the standard deviation and other statistics.

What measure of variability is the average square distance of the scores from the mean?

The variance is the average squared distance to the mean. However, it is erroneous to think that the standard deviation (the square root of the variance) equals the average distance to the mean.