normal distribution sampling and probability. properties of a normal distribution mean = median =...
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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)
Age of Respondent
90.0
85.0
80.0
75.0
70.0
65.0
60.0
55.0
50.0
45.0
40.0
35.0
30.0
25.0
20.0
300
200
100
0
Std. Dev = 17.81
Mean = 45.6
N = 1514.00
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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
HIGHEST YEAR OF SCHOOL COMPLETED
20.018.016.014.012.010.08.06.04.0
600
500
400
300
200
100
0
Std. Dev = 2.84
Mean = 13.9
N = 1845.00
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Information about previous graph
Mean 2.845
Standard Deviation 13.93
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
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
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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 = 23Mean = 25SD = 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 = 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
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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