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 75 th and 25 th percentiles and dividing by 2. Shortcomings –Does not allow for precise interpretation of a score within a distribution –Not used for inferential statistics.

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.

Variance (second moment about the mean) The Variance, s 2, 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 The Variance, s 2, is the sample variance, and is used to estimate the actual population variance,  2

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

Calculating the Standard Deviation

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 = s = 1.25 The same test given to second- through fourth- grade children. 2, 8, 9, 11, 15, 17, 20 xbar = 11.71s = 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

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

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

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

Transformation scores We can transform scores to have a mean and standard deviation of our choice. Why might we want to do this?

With our scores We want: –Mean = 100 –s = 15 Transform: –8 & 10.

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 ).

Examples of Standard Scores