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Normal Distribution Sampling and Probability

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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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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

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Skewed Distribution (positive skew)

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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.

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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.

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Negative Skew

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Information about previous graph Mean2.845 Standard Deviation13.93 Skew-.084 Kurtosis.252

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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.

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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.

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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.

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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.

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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.

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One standard deviation:

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Two standard deviations from the mean:

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Three standard deviations from the mean:

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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.

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Determining what scores fall within 1, 2, and 3 SD from the Mean Mean = 25 SD = 4 AboveBelow 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

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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.

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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!

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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%

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But where does this client fall in the distribution compared to other clients. What percent received higher scores? What percent received lower scores?

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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

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Location of Z scores

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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.

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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

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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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