#### Transcript Measures of Dispersion & The Standard Normal Distribution

Measures of Dispersion & The Standard Normal Distribution 9/12/06 The Semi-Interquartile Range (SIR) • A measure of dispersion obtained by finding the difference between the 75th and 25th percentiles and dividing by 2. • Shortcomings – Does not allow for precise interpretation of a score within a distribution – Not used for inferential statistics. Q1 Q1 SIR 2 Calculate the SIR 6, 7, 8, 9, 9, 9, 10, 11, 12 • Remember the steps for finding quartiles – First, order the scores from least to greatest. – Second, Add 1 to the sample size. – Third, Multiply sample size by percentile to find location. – Q1 = (10 + 1) * .25 – Q2 = (10 + 1) * .50 – Q3 = (10 + 1) * .75 » If the value obtained is a fraction take the average of the two adjacent X values. Q1 Q1 SIR 2 Variance (second moment about the mean) • The Variance, s2, represents the amount of variability of the data relative to their mean • As shown below, the variance is the “average” of the squared deviations of the observations about their mean s 2 ( x x) 2 i n 1 • The Variance, s2, is the sample variance, and is used to estimate the actual population variance, s 2 s 2 2 ( x ) i N Standard Deviation • Considered the most useful index of variability. • It is a single number that represents the spread of a distribution. • If a distribution is normal, then the mean plus or minus 3 SD will encompass about 99% of all scores in the distribution. Definitional vs. Computational • Definitional – An equation that defines a measure • Computational – An equation that simplifies the calculation of the measure s 2 ( x x) 2 i n 1 ( X ) X 2 N s n 1 2 2 Calculate the variance using the computational and definitional formulas. • 6, 7, 8, 9, 9, 9, 10, 11, 12 s 2 ( x x) i n 1 2 ( X ) X 2 n s n 1 2 2 Calculating the Standard Deviation s s 2 • Interpreting the standard deviation – Remember • Fifty Percent of All Scores in a Normal Curve Fall on Each Side of the Mean Probabilities Under the Normal Curve With our previous scores • What score is one standard deviation above the mean? – Two standard deviations? – Three standard deviations? • What score is one standard deviation below the mean? – Two standard deviations? – Three standard deviations? Interpreting the standard deviation • We can compare the standard deviations of different samples to determine which has the greatest dispersion. – Example • A spelling test given to third-grader children 10, 12, 12, 12, 13, 13, 14 xbar = 12.28 s = 1.25 • The same test given to second- through fourthgrade children. 2, 8, 9, 11, 15, 17, 20 xbar = 11.71 s = 6.10 The shape of distributions • Skew – A statistic that describes the degree of skew for a distribution. • 0 = no skew • + or - .50 is sufficiently symmetrical 3 ( x Median ) 3 s s ^ Kurtosis • Mesokurtic (normal) – Around 3.00 • Platykurtic (flat) – Less than 3.00 • Leptokurtic (peaked) – Greater than 3.00 Q3 Q1 s 3 2( P90 P10 ) ^ 4 From our previous scores • Calculate the skew 6, 7, 8, 9, 9, 9, 10, 11, 12 3 ( x Median ) 3 s s ^ xbar = 9.00 mdn = 9.00 s = 1.87 • Calculate Kurtosis 6, 7, 8, 9, 9, 9, 10, 11, 12 Q3 =10.5 Q1 = 7.5 P10 = 6 P90 = 12 Q3 Q1 s 3 2( P90 P10 ) ^ 4 The Standard Normal Distribution • Z-scores – A descriptive statistic that represents the distance between an observed score and the mean relative to the standard deviation xx z s X z s Standard Normal Distribution • Z-scores – Convert and distribution to: • Have a mean = 0 • Have standard deviation = 1 – However, if the parent distribution is not normal the calculated z-scores will not be normally distributed. Why do we calculate z-scores? • To compare two different measures – e.g., Math score to reading score, weight to height. – Area under the curve • Can be used to calculate what proportion of scores are between different scores or to calculate what proportion of scores are greater than or less than a particular score. Class practice 6, 7, 8, 9, 9, 9, 10, 11, 12 Calculate z-scores for 8, 10, & 11. What percentage of scores are greater than 10? What percentage are less than 8? What percentage are between 8 and 10? Z-scores to raw scores • If we want to know what the raw score of a score at a specific %tile is we calculate the raw using this formula. x z (s) x Transformation scores • We can transform scores to have a mean and standard deviation of our choice. • Why might we want to do this? x z (s) x With our scores • We want: – Mean = 100 – s = 15 • Transform: – 8 & 10. x z (s) x