Sections 15-17 Standard Deviation (SD)

Question 1

Standard Deviation

Answer 1

STANDARD DEVIATION is the most used form of VARIABILITY in statistics.

* It is a measure of HOW MUCH SCORES DIFFER (or vary) from the MEAN of the SCORES.

* It is IMPORTANT because it gives a BETTER view of how the data is distributed.

* Symbols for STANDARD DEVIATION include S, σ (the lowercase Greek letter Sigma), S.D., or SD.

Let’s go through the Example below:

Step 1) Look at the Formula for Standard Deviation, but don’t let that scare you. Just take it one step at a time.

Step 2) First, we are told what X equals in the example. X is the SCORES that we’re working with – 6 of them to be exact, and listed from lowest to highest.

Step 3) In the next column we did two things.

• We calculated the MEAN of the scores (10+11+11+13+14+19) / 6 = 13
• And we subtracted the mean from EACH of the Scores to get the DEVIATION (NOT the Standard Deviation, though) Ex: The first Score (10) DEVIATES -3 from the mean of all the scores.

Step 4) in the final column, we also did two things:

• We SQUARED the DEVIATIONS from the prior column. (Ex: In the first row -3 x -3 = 9.00)
• Then we ADDED (or summed) all of those squared deviations to come up with a SUM of the DEVIATIONS SQUARED = 54.00

Step 5) Finally, we INSERT all these VALUES back INTO the original FORMULA for STANDARD DEVIATION (Remember: N = number of observations (or scores)

* And don’t sweat the formula. You can take that with you into the EXAM. You’re not expected to memorize that, just remember HOW to use it.

Question 2

Standard Deviation, Normal Distribution, and the 68% Rule

Answer 2

• So it makes sense that the larger the deviations from the mean, the larger the standard deviation. Conversely, the smaller the deviations from the mean, the smaller the standard deviation.
• If all the scores were the same, the standard deviation would equal zero because the mean would be equal to all the scores and so there would be no deviation from the mean.
• Remember that the STANDARD DEVIATION is a first cousin of the MEAN. So with the mean (the most popular average), they usually also report the standard deviation. (Just as the MEDIAN usually goes with the INTERQUARTILE RANGE)

This next point is VERY IMPORTANT:

• The STANDARD DEVIATION takes on a special meaning when considered in relation to the NORMAL CURVE (the curve of a Normal Distribution) because it was designed specifically to describe this distribution.
• Here is a SIMPLE RULE to remember: About two-thirds of the cases lie within one standard deviation unit of the mean in a normal distribution. (Note that “within one standard deviation unit” means one unit on both sides of the mean.) For instance, suppose that the mean of a set of normally distributed scores equals 70.00 and the standard deviation equals 10.00. About two-thirds of the cases lie within 10 points of the mean.
• More precisely, 68% of the cases lie within 10 points of the mean, as illustrated in Figure 1 on the next page

Question 3

The 95% Rule and the 99.7% Rule

Answer 3

68% RULE – 68% of the scores lie within ONE STANDARD DEVIATION from the MEAN in a NORMAL DISTRIBUTION

95% RULE – 95% of the scores lie within TWO STANDARD DEVIATION from the MEAN in a NORMAL DISTRIBUTION

99.7% RULE – 99.7% of the scores lie within THREE STANDARD DEVIATION from the MEAN in a NORMAL DISTRIBUTION

• NOTE: the 95% RULE and the 99.7% RULE are APPROXIMATE.

Example: (Refer to figures 1. and 2. below) Imagine the MEAN (M) for a group equals 35.00 and the standard deviation equals 6.00.

To ILLUSTRATE the 95% RULE:

• TWO standard deviation units equal 12.00 points (2 x 6.00 = 12.00).
• Thus, if you go up 12 points from the mean (35.00 + 12.00 = 47.00) and go down 12 points from the mean (35.00- 12.00 = 23.00), you have identified the scores (47.00 and 23.00) between which approximately 95% of the cases lie.

To ILLUSTRATE the 99.7% RULE:

• THREE standard deviation units equal 18.00 points (3 x 6.00 = 18.00).
• Thus, if you go up 18 points from the mean (35.00 + 18.00 = 53.00) and go down 18 points from the mean (35.00 - 18.00 = 17.00), you have identified the scores (53.00 and 17.00) between which approximately 99.7% of the cases lie.

In review, If M = 35.00 and S = 6.00, then:

(1) 68% of the cases lie between 29.00 and 41.00
(2) 95% of the cases lie between 23.00 and 47.00
(3) 99.7% ofthe cases lie between 17.00 and 53.00.

Question 4

How to Compare Distributions with Different Standard Deviations

Answer 4

If you have two distributions with the SAME MEAN but different STANDARD DEVIATIONS, the distribution with the LARGER STANDARD DEVIATION will have:

• a LARGER VARIABILITY of SCORES
• The SHAPE of the NORMAL CURVE (The BELL CURVE) will be WIDER than the shape of another NORMAL CURVE with a SMALLER STANDARD DEVIATION.

Question 5

PRECISE 95% and 99% rules

Answer 5

In a NORMAL DISTRIBUTION:

• 68% of scores are within 1 SD (Standard Deviation)
• Approximately 95% of scores are within 2 SD
• Approximately 99.7% of scores are within 3 SD

Now let’s be more PRECISE instead of APPROXIMATE.

PRECISE 95% says that if you go out 1.96 STANDARD DEVIATIONS from the mean in a NORMAL distribution, you will find 95% of the cases.

PRECISE 99% says that if you go out 2.58 STANDARD DEVIATIONS from the mean in a NORMAL distribution, you will find 99% of the cases.

Example: If a set of scores had a MEAN of 44.00 and a STANDARD DEVIATION of 4.00, then…

• 68% of the scores would be within 1 SD of the MEAN
  
  • Between 40.00 and 48.00
    
    • (44.00 - 4.00 = 40.00, 44.00 + 4 = 48.00)
• 95% of the scores would be within 1.96 SD of the MEAN
  
  • First, calculate 1.96 SD = 1.96 x 4 = 7.84, so…
  • Between 36.16 and 51.84
    
    • (44.00 - 7.84 = 36.16, 44.00 + 7.84 = 51.84)
• 99% of the scores would be within 2.58 SD of the MEAN
  
  • First, calculate 2.58 SD = 2.58 x 4 = 10.32, so…
  • Between 33.68 and 54.32
    
    • (44.00 - 10.32 = 33.68, 44.00 + 10.32 = 54.32)
• So 68% of scores would lie between 40.00 and 48.00
• 95% of scores would lie between 36.16 and 51.84
• And 99% of scores would lie between 33.68 and 54.32

…PRECISELY!