Measures of Dispersion

DR.VHEMALATHA

ASSITANT PROFESSOR OF MATHS CPA BODI

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Definition

• Measures of dispersion are descriptive statistics that describe how similar a set of scores are to each other
• The more similar the scores are to each other, the lower the measure of dispersion will be
• The less similar the scores are to each other, the higher the measure of dispersion will be
• In general, the more spread out a distribution is, the larger the measure of dispersion will be

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Measures of Dispersion

• Which of the distributions of scores has the larger dispersion?

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• The upper distribution has more dispersion because the scores are more spread out
• That is, they are less similar to each other

Measures of Dispersion

• There are three main measures of dispersion:
• The range
• The semi-interquartile range (SIR)
• Variance / standard deviation

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The Range

• The range is defined as the difference between the largest score in the set of data and the smallest score in the set of data, X_L - X_S
• What is the range of the following data:�4 8 1 6 6 2 9 3 6 9
• The largest score (X_L) is 9; the smallest score (X_S) is 1; the range is X_L - X_S = 9 - 1 = 8

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When To Use the Range

• The range is used when
• you have ordinal data or
• you are presenting your results to people with little or no knowledge of statistics
• The range is rarely used in scientific work as it is fairly insensitive
• It depends on only two scores in the set of data, X_L and X_S
• Two very different sets of data can have the same range:�1 1 1 1 9 vs 1 3 5 7 9

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The Semi-Interquartile Range

• The semi-interquartile range (or SIR) is defined as the difference of the first and third quartiles divided by two
• The first quartile is the 25^th percentile
• The third quartile is the 75^th percentile
• SIR = (Q₃ - Q₁) / 2

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SIR Example

• What is the SIR for the data to the right?
• 25 % of the scores are below 5
• 5 is the first quartile
• 25 % of the scores are above 25
• 25 is the third quartile
• SIR = (Q₃ - Q₁) / 2 = (25 - 5) / 2 = 10

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When To Use the SIR

• The SIR is often used with skewed data as it is insensitive to the extreme scores

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Variance

• Variance is defined as the average of the square deviations:

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What Does the Variance Formula Mean?

• First, it says to subtract the mean from each of the scores
• This difference is called a deviate or a deviation score
• The deviate tells us how far a given score is from the typical, or average, score
• Thus, the deviate is a measure of dispersion for a given score

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What Does the Variance Formula Mean?

• Why can’t we simply take the average of the deviates? That is, why isn’t variance defined as:

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What Does the Variance Formula Mean?

• One of the definitions of the mean was that it always made the sum of the scores minus the mean equal to 0
• Thus, the average of the deviates must be 0 since the sum of the deviates must equal 0
• To avoid this problem, statisticians square the deviate score prior to averaging them
• Squaring the deviate score makes all the squared scores positive

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What Does the Variance Formula Mean?

• Variance is the mean of the squared deviation scores
• The larger the variance is, the more the scores deviate, on average, away from the mean
• The smaller the variance is, the less the scores deviate, on average, from the mean

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Standard Deviation

• When the deviate scores are squared in variance, their unit of measure is squared as well
• E.g. If people’s weights are measured in pounds, then the variance of the weights would be expressed in pounds² (or squared pounds)
• Since squared units of measure are often awkward to deal with, the square root of variance is often used instead
• The standard deviation is the square root of variance

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Standard Deviation

• Standard deviation = √variance
• Variance = standard deviation²

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Computational Formula

• When calculating variance, it is often easier to use a computational formula which is algebraically equivalent to the definitional formula:

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• σ² is the population variance, X is a score, μ is the population mean, and N is the number of scores

Computational Formula Example

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Computational Formula Example

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Variance of a Sample

• Because the sample mean is not a perfect estimate of the population mean, the formula for the variance of a sample is slightly different from the formula for the variance of a population:

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• s² is the sample variance, X is a score, X is the sample mean, and N is the number of scores

Measure of Skew

• Skew is a measure of symmetry in the distribution of scores

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Measure of Skew

• The following formula can be used to determine skew:

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Measure of Skew

• If s³ < 0, then the distribution has a negative skew
• If s³ > 0 then the distribution has a positive skew
• If s³ = 0 then the distribution is symmetrical
• The more different s³ is from 0, the greater the skew in the distribution

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Kurtosis�(Not Related to Halitosis)

• Kurtosis measures whether the scores are spread out more or less than they would be in a normal (Gaussian) distribution

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Kurtosis

• When the distribution is normally distributed, its kurtosis equals 3 and it is said to be mesokurtic
• When the distribution is less spread out than normal, its kurtosis is greater than 3 and it is said to be leptokurtic
• When the distribution is more spread out than normal, its kurtosis is less than 3 and it is said to be platykurtic

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Measure of Kurtosis

• The measure of kurtosis is given by:

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s², s³, & s⁴

• Collectively, the variance (s²), skew (s³), and kurtosis (s⁴) describe the shape of the distribution

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