10_week_1_session_2_final_copy_summ08-1 - Statistics 10...

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Statistics 10 Summer 08 Professor Esfandiari Week 1 – Session 2 The objective of this lecture is to: Introduce you to the measures of center and when to use each one Introduce you to the measures of spread and when to use each one Show you how to read and interpret outputs for measures of center and spread Show you how to use measures of center and spread to compare the two boxplots Measures of Center Mean = the average of scores. It is the most stable measure of center. Mean or Y = Y/ N Median = the score corresponding to the 50 th percentile. It is not effected by outliers. Appropriate measure of center for skewed distributions. Mode = the score with the highest frequency . It is the least stable measure of center. The three measures of center are equal for symmetrical distributions. Outliers or extreme scores impact the mean and not the median. This impact is much higher when the sample size is small. Measures of Spread Range = maximum - minimum Inter-quartile range = upper quartile – lower quartile 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. 1
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Y (Y - Y ) (Y - Y )^2 3 -5 25 5 -3 9 8 0 0 10 +2 4 14 +6 36 Mean or Y = (3 + 5 + 8 + 10 + 15)/5 = 8 Variance or S^2 = S Y 2 = ( Y - Y ) 2 N - 1 = (25 + 9 + 4 + 36) / 4 = 74/4 = 18.5 S or Standard Deviation = square root of 18.5 = 4.30 Exercise to be done in lecture Mark was computing the variance of 50 observations and he found (Y - Y ) to be zero. Jack who was computing the variance for 500 observations found (Y - Y ) to be zero. Is this because Jack’s sample was so much larger? What reason can you think of? Create a data set with six observations for which variance is equal to zero. Draw a figure (histogram) for which standard deviation is equal to zero. Shapes of histograms Histograms represent the frequency distribution for quantitative variables and they can be either symmetrical or skewed. In the case of symmetrical distributions, the mean and the median will are very close to each other. But, the scatter of the points around the center can be different. Some symmetrical histograms will have a higher concentration of the data around the mean and some will have a lower concentration of the data around the mean. In other words, in the case of some histograms the data are more spread out.
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This note was uploaded on 08/15/2008 for the course STATS 10 taught by Professor Ioudina during the Summer '08 term at UCLA.

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10_week_1_session_2_final_copy_summ08-1 - Statistics 10...

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