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.

213 QQ

SIR

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.

213 QQ

SIR

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

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• The Variance, s2, is the sample variance, and is used to estimate the actual population variance, 2

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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

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Calculating the Standard Deviation

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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 children10, 12, 12, 12, 13, 13, 14xbar = 12.28 s = 1.25

• The same test given to second- through fourth-grade children.

2, 8, 9, 11, 15, 17, 20xbar = 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

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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

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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

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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

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Transformation scores

• We can transform scores to have a mean and standard deviation of our choice.

• Why might we want to do this?

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With our scores

• We want:– Mean = 100– s = 15

• Transform:– 8 & 10.

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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