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Lecture5:Measures of DispersionBy: Amani Albraikan

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Measures of Dispersion Synonym for variability Often called “spread” or “scatter” Indicator of consistency among a data

set Indicates how close data are clustered

about a measure of central tendency Commonly used measures of dispersion

◦ Range◦ Interquartile Range◦ Standard Deviation

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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

Definition

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Measures of Dispersion Which of the

distributions of scores has the larger dispersion? 0

255075

100125

1 2 3 4 5 6 7 8 9 10

0255075

100125

1 2 3 4 5 6 7 8 9 10

The upper distribution has more dispersion because the scores are more spread out

That is, they are less similar to each other

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There are three main measures of dispersion:◦ The range (المدى)◦ The semi-interquartile range (SIR) الربيعي المدى نص7ف◦ Variance / standard deviation( االنحراف و التباين

(المعياري

Measures of Dispersion

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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, XL - XS

What is the range of the following data:4 8 1 6 6 2 9 3 6 9

The largest score (XL) is 9; the smallest score (XS) is 1; the range is XL - XS = 9 - 1 = 8

The Range

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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,

XL and XS◦ Two very different sets of data can have the same

range:1 1 1 1 9 vs 1 3 5 7 9

When To Use the Range

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Used for Interval-Ratio Level Variables Difference between the Highest and Lowest

Values. Range= Maximum value – minimum value Often expressed as the full equation (i.e.,

There was a 19 point range, with a maximum score of 96 and a minimum of 77.)

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Represents the width of the middle 50% of the data.

The 3rd quartile (Q3 ; 75th percentile) minus the 1st quartile (Q1; 25th percentile) equals the interquartile range (IQR).

Formula: IQR=Q3-Q1

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Order the values of an interval-ratio level variable from lowest to highest.

Multiple N by .75 to find where Q3 is located and by .25 to find Q1. If Q3 and Q1 are not even numbers, find the values on either side and average them.

Subtract the value at Q1 from the one at Q3.

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68, 72, 73, 76, 79, 79, 82, 83, 84, 84, 86, 88, 90, 91, 91, 95, 96, 97, 99, 99

N=20 Q1=20 * .25= 5th case=79 Q3=20 * .75=15th case=91 91-79=12 Interpretation: The middle 50% of data falls

between 79 and 91, with an inter-quartile range of 12 points.

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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 25th percentile◦ The third quartile is the 75th percentile

SIR = (Q3 - Q1) / 2

The Semi-Interquartile Range

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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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5 = 25th %tile

81012142030

25 = 75th %tile

60

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

When To Use the SIR

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Variance: Used only for interval-ratio level variables. Assesses the average squared deviations of all values from the mean value.

Standard Deviation: The square root of the variance. Roughly estimated to be the average amount of deviation between each score and the mean.

Deviation from the mean

Variance Variance for a population is symbolized by:

2

Variance for a sample is s2

Formula:

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Variance of a Sample

1NXX

s2

2

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

Steps to Calculate Variance Calculate the mean Subtract the mean from each value (x-x) Square each deviation from the mean (x-

x)2

Sum the squared deviations Divide by N-1

Variance Example Values= 1, 5, 8,

38, 39, 42 Mean=22 N=6 (x-x)2=1871 1871/ N-1 1871/ 5 = 374.2 S2=374.2

Value x-x x-x2

1 1-22=-21 441

5 5-22=-17 289

8 8-22=-14 196

38 38-22=16 256

39 39-22=17 289

42 42-22=20 400

1871

Calculate the Variance and Standard Deviation Values:

◦ 12, 15, 18, 21, 22, 23, 26, 32

How to Recode a Variable in SPSS Select “Transform” Select “Recode into different variables” Select variable you wish to recode and click on

arrow to send it into box on right Give your new variable a name and a label and

click “change” Select “old and new values” For each original variable value enter the old

value and the new value and click “add” Decide how you want to handle missing data,

etc… Click “continue” Click “okay”

Recode the variable “Opinion of Family Income” Select: Transform, Recode into new variable Send variable into center box Create new name and label: “finrecode” &

“Opinion of family income 3 categories” Old values: 1-2=1; 3=2, 4-5=3,

missing=missing Continue Okay Add new value labels to new variable:

◦ 1=below average, 2=average, 3=above average

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Variance is defined as the average of the square deviations:

Variance

1NXX

s2

2

1NXX

s2

2

أو

fi

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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

What Does the Variance Formula Mean?

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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

What Does the Variance Formula Mean?

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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

Standard Deviation

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Standard deviation = variance Variance = standard deviation2

Standard Deviation

SPSS Exercise Open Data File: GSS02PFP-B.sav Calculate the IQV of the variable class Calculate the Range, IQR, Variance and Standard Deviation

of the variables: age, childs, educ, hrs1 Make a box plot with educ and class Make a box plot with hrs1 and class Recode hrs1 into hrs2: Convert interval-ratio variable into

ordinal variable with 4 categories: 0= 0 Didn’t work, 1-32=1 part-time, 33-45=2 full time, 45 to highest value=3 more than full-time

Make a simple bar graph with the new variable hrs2 Make a clustered bar graph with the new variable hrs2 and

another categorical variable (select % instead of count)