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10_week_2_session_1_summer_08

# 10_week_2_session_1_summer_08 - 1 Statistics 10 Professor...

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Statistics 10 Professor Esfandiari Summer2008 Week 2 session 1 The objective of this handout is to: Review the concept of standard deviation, Discuss the effects of linear transformation on mean, variance, and standard deviation, Introduce you to standardization as an example of linear transformation, Show you how standard scores (Z scores) enable you to compare scores that were measured with different units, Introduce you to the normal model and the standard normal curve, Show you how to calculate and interpret Z, Show you how to transform percentile ranks into Z scores and Z scores into raw scores, Show you how to check for normality through the 68%, 95%, and 99.7% rule, Show you how to check for normality through the pplot, Discuss Pplot for normal and skewed data, and Warn you not to use the normal approximation for distributions that are not normal. Variance = the sum of the square of the average deviations from the mean divided by N – 1: S Y 2 = ( Y - Y ) 2 N - 1 You subtract each score from the mean (Y – Y ) The ∑(Y – Y ) = 0; some scores are higher and some lower than the mean and the sum is zero To make the sum non-zero, you square it To make the sum independent of sample size , you divide by N -1. In the future discussions, we will explain why we divide by N- 1 rather than N. Standard deviation = The square root of variance The measure of scatter needs to be in terms of the same unit as X, since we squared the deviations around the mean to find the variance, we will take the square root of the variance to find standard deviation which is the measure of spread. Linear transformation could be one of the following or a combination of them: Adding a constant to all the scores, Subtracting all the scores from a constant, Multiplying all the scores by a constant, and Dividing all the scores by a constant. 1

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Table 1. In the following table we will summarize the effect of the above transformations on the mean, variance, and standard deviation, and then take a practical example: Linear transformation Effect on the mean Or Xbar Effect on variance Or S square Effect on standard deviation or S Adding a constant to all the scores The constant will be added to the mean. Variance will not change. SD will not change. Subtracting all the scores from a constant. The constant will be subtracted from the mean. Variance will not change. SD will not change. Multiplying all the scores by a constant The mean will be multiplied by the constant. Variance will be multiplied by the square of the constant. SD will be multiplied by the absolute value of the constant. Dividing all the scores by a constant The mean will be divided by the constant. Variance will be divided by the square of the constant.
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10_week_2_session_1_summer_08 - 1 Statistics 10 Professor...

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