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Measures of Dispersion & The Standard Normal Distribution 2/5/07 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. Q3 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. Q3 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) i n 1 2 SS 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 SS N Standard Deviation • Considered the most useful index of variability. – Can be interpreted in terms of the original metric • 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) s2 2 i n 1 n X 2 ( X ) 2 n(n 1) Calculating the Standard Deviation s s 2 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 • 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 The shape of distributions • Skew – A statistic that describes the degree of skew for a distribution. • 0 = no skew – + or - .50 is sufficiently symmetrical • + value = + skew • - value = - skew • You are not expected to calculate by hand. – Be able to interpret 3 ( X X ) N 3 s ( X X ) 1.5 [ ] N Kurtosis • Mesokurtic (normal) – Around 3.00 • Platykurtic (flat) – Less than 3.00 • Leptokurtic (peaked) – Greater than 3.00 • You are not expected to calculate by hand. – Be able to interpret 4 ( X X ) N 4 2 s ( X X ) [ ]2 N 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 xi x z s z xi s Standard Normal Distribution • Z-scores – Convert a 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. • Used to set cut score for screening instruments. 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. • With previous scores what is the raw score – 90%tile – 60%tile – 15%tile 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 Key points about Standard Scores • Standard scores use a common scale to indicate how an individual compares to other individuals in a group. • The simplest form of a standard score is a Z score. • A Z score expresses how far a raw score is from the mean in standard deviation units. • Standard scores provide a better basis for comparing performance on different measures than do raw scores. • A Probability is a percent stated in decimal form and refers to the likelihood of an event occurring. • T scores are z scores expressed in a different form (z score x 10 + 50). Examples of Standard Scores