Measures of Dispersion&

The Standard Normal Distribution

9/13/06

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.

213 QQSIR

Calculate the SIR6, 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 QQSIR

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

1)( 2

2

n

xxs i

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

Nxi

2

2 )(

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

1

)( 22

2

nNX

Xs

1)( 2

2

n

xxs i

Calculate the variance using the computational and definitional

formulas.• 6, 7, 8, 9, 9, 9, 10, 11, 12

1)( 2

2

n

xxs i

1

)( 22

2

nnX

Xs

Calculating the Standard Deviation

2ss

• 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 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

The shape of distributions

• Skew– A statistic that

describes the degree of skew for a distribution.

• 0 = no skew• + or - .50 is sufficiently

symmetrical

sMedianx

s)(33

^

Kurtosis

• Mesokurtic (normal)– Around 3.00

• Platykurtic (flat)– Less than 3.00

• Leptokurtic (peaked)– Greater than 3.00 )(2

31090

13^4

PPQQs

From our previous scores• Calculate the skew6, 7, 8, 9, 9, 9, 10, 11,

12

xbar = 9.00mdn = 9.00 s = 1.87

sMedianx

s)(33

^

• Calculate Kurtosis6, 7, 8, 9, 9, 9, 10, 11,

12Q3 =10.5Q1 = 7.5 P10 = 6P90 = 12

)(23

1090

13^4

PPQQs

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

sxxz

Xz

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. xszx )(

Transformation scores

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

• Why might we want to do this?

xszx )(

With our scores

• We want:– Mean = 100– s = 15

• Transform:– 8 & 10.

xszx )(