Normal Distribution

Sampling and Probability

Properties of a Normal Distribution

• Mean = median = mode

• There are the same number of scores below and above the mean.

• 50% of the scores are on either side of the mean.

• The area under the normal curve totals 100%..

Not all distributions are bell-shaped curves

• Some distributions lean to the right

• Some distributions lean to the left

• Some distributions are tall or peaked

• Some distributions are flat on top

Skewed Distribution (positive skew)

Age of Respondent

90.0

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55.0

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45.0

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Std. Dev = 17.81

Mean = 45.6

N = 1514.00

Things to remember about skewed distributions

• Most distributions are not bell-shaped.

• Distributions can be skewed to the right or the left.

• The direction of the skew pertains to the direction of the tail (smaller end of the distribution).

• Tails on the right are positively skewed.

• Tails on the left are negatively skewed.

Other things to know about skewness

• Skewness is a measure of how evenly scores in a distribution are distributed around the mean.

• The bigger the skew the larger is the degree to which most scores lie on one side of the mean versus the other side.

• Skew scores produced in SPSS are either negative or positive, indicating the direction of the skew.

Negative Skew

HIGHEST YEAR OF SCHOOL COMPLETED

20.018.016.014.012.010.08.06.04.0

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Std. Dev = 2.84

Mean = 13.9

N = 1845.00

Information about previous graph

Mean 2.845

Standard Deviation 13.93

Skew -.084

Kurtosis .252

Knowing whether a distribution is skewed:

• Tells you if you have a normal distribution. (If the skew is close to zero, you may have a normal distribution)

• Tells you whether you should use mean or median as a measure of central tendency. (The greater the skew, the more likely that the median is the better measure of central tendency)

• Tells you whether you can use inferential statistics.

Another measure of the shape of the distribution is

kurtosis. This measure tells you:

• Whether the standard deviation (degree to which scores vary from the mean) are large or small)

• Curves that are very narrow have small standard deviations.

• Curves that are very wide have large standard deviations.

• A kurtosis measure that is near zero (less than 1) may indicate a normal distribution.

Other Important Characteristics of a Normal Distribution

• The shape of the curve changes depending on the mean and standard deviation of the distribution.

• The area under the curve is 100%

• A mathematical theory, the Central Limit Theorem, allows us to determine what scores in the distribution are between 1, 2, and 3 standard deviations from the mean.

Central Limit Theorem:

• 68.25% of the scores are within one standard deviation of the mean.

• 94.44% of the scores are within two standard deviations of the mean.

• 99.74%(or most of the scores) are within 3 standard deviations of the mean.

Central Limit Theorem also means that:

• 34.13% of the scores are within one standard deviation above or below the mean.

• 47.12% of the scores are within two standard deviations above or below the mean.

• 49.87% of the scores are within three standard deviations above or below the mean.

One standard deviation:

Two standard deviations from the mean:

Three standard deviations from the mean:

This theory allows us to:

• Determine the percentage of scores that fall within any two scores in a normal distribution.

• Determine what scores fall within one, two, and three standard deviations from the mean.

• Determine how a score is related to other scores in the distribution (What percentage of scores are above or below this number).

• Compare scores in different normal distributions that have different means and standard deviations.

• Estimate the probability with which a number occurs.

Determining what scores fall within 1, 2, and 3 SD from the Mean

Mean = 25

SD = 4

Above Below

1 SD = 25 + 4 = 29 = 25 – 4 = 21

2 SD = 25 + (2 * 4) = 25 + 8 = 33

= 25 – (2 * 4) =

25 – 8 = 17

3 SD = 25 + (3 * 4) = 25 + 12 = 37

25 – (3 * 4) = 25 – 12 = 13

To do some of these things we need to convert a specific raw score to a z score. A z score is:

• A measure of where the raw score falls in a normal curve.

• It allows us to determine what percentage of scores are above or below the raw score.

The formal for a z score is:

(Raw score – Mean) Standard Deviation

If the raw score is larger than the mean, the z score will be positive. If the raw score is smaller than the mean, the z score will be negative. This means that when we try to compare the raw score to the distribution, a positive score will be above the mean and a negative score will be below the mean!

For example, (Assessment of Client Economic Hardship)

Raw Score = 23Mean = 25SD = 1.5

23-25 - 2 = - 1.33 1.5 1.5

Percent of Area Under Curve = 40.82%

But where does this client fall in the distribution compared to other clients. What percent

received higher scores?What percent received lower

scores?

To convert z scores to percentages;

• Use the chart (Table 8.1) on p. 132 of your textbook (Z charts can be found in any statistics book).

• Area under curve for a z score of -1.33 is 40.82

Location of Z scores

To find out how many clients had lower or

higher economic hardship scores:

Lower Scores – Area of curve below the mean = 50%

50% – 40.82% = 9.18%

Consequently 9.18% had lower scores.

Higher scores = 40.82% + 50% (above the mean) = 90.82% had higher scores.

Positive Z score example:

Raw Score = 15Mean = 12Standard Deviation = 5

= 15 – 12 = 3 = .60 = 22.57 5 5 50 – 22.57 = 27.43% had higher scores on the

economic hardship scale50 + 22.57 = 72.57 had lower scores on the scale

How do scores from different distributions compare:

Distribution 1

Raw Score = 10 Mean = 5 SD = 2

= 10 – 5 = 5 = 2.5 = 49.38 = .62% above

2 2 99.38% below

Distribution 2

Raw Score= 9 Mean = 4 SD = 2.2

= 9 - 4 = 5 = 2.27 = 48.84 = 1.16% above

2.2 2.2 98.84% below