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Lecture5: Measures of Dispersion. By: Amani Albraikan. 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

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## Lecture5: Measures of Dispersion

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**Lecture5:Measures of Dispersion**By: Amani Albraikan**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**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**Measures of Dispersion**• Which of the distributions of scores has the larger dispersion? • 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 (المدى) • The semi-interquartile range (SIR)نصف المدى الربيعي • Variance / standard deviation(التباين و الانحراف المعياري)**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, 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**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, 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**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.)**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**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.**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.**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 25th percentile • The third quartile is the 75th percentile • SIR = (Q3 - Q1) / 2**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**When To Use the SIR**• The SIR is often used with skewed data as it is insensitive to the extreme scores**Deviation from the mean**• 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.**Variance**• Variance for a population is symbolized by: 2 • Variance for a sample is s2 Formula:**Variance of a Sample**• 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**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**Variance**• Variance is defined as the average of the square deviations: أو fi**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**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**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**Standard Deviation**• Standard deviation = variance • Variance = standard deviation2**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)

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