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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%.

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Normal Distribution

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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)


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


Information about previous graph


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


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 = 23

Mean = 25

SD = 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 = 15

Mean = 12

Standard Deviation = 5

= 15 – 12 = 3 = .60 = 22.57

5 5

50 – 22.57 = 27.43% had higher scores on the economic hardship scale

50 + 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


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