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  • 00:01

    DANIEL LITTLE: In this video, I will describethree measures that tell us about the variabilityof a distribution, specifically I'lltalk about the range, the standard deviationin the variance, and the sum of squares.I realize I've just said that I will talk about three measures,but I've actually given you four.The reason for this is because twoof the measures, standard deviation and variance,

  • 00:22

    DANIEL LITTLE [continued]: are actually quite closely related to one another.In fact, both standard deviation and varianceare very much related to the sum of squares.So first I'll talk about the range,and then I will talk about all of the remaining threetogether.Let's start by assuming that we havea normal distribution from which we sample or observe some data.

  • 00:44

    DANIEL LITTLE [continued]: Here we can see our data being generatedby this normal distribution.Each one of those points represents a particular valuethat we might observe in our data.For this particular example, I'vespecified that the normal distribution has a mean of 100

  • 01:04

    DANIEL LITTLE [continued]: and a standard deviation of 10.You could think of these as being representativeof the types of data values you mightobserve if you were looking at IQ scores, for instance.Now typically, we don't have access to our distribution.In fact, all we have access to is the listof values which we recorded.

  • 01:25

    DANIEL LITTLE [continued]: Here I've taken each of these valuesand laid them out in a table on the left-hand sideof the screen.Now, the first measure we can compute is called the range.And the range indicates the spread of scoresthat we might observe in our data.To find the range, all we would dois simply take the highest value-- here it is 127--

  • 01:50

    DANIEL LITTLE [continued]: and subtract from that our lowest value of 81.In this particular example, if we do the subtraction,we find that it equals 46.Now, there are other types of rangesthat we can also compute.

  • 02:11

    DANIEL LITTLE [continued]: We don't need to take just the highest scoreand subtract off the lowest score.We can compute things like the interquartile range, whichwould require us to find the point at which 25% of our datais greater than, which might be somewhere here.And the point at which 75% of our data

  • 02:32

    DANIEL LITTLE [continued]: is less than, which might be a point here.In which case we would compute the interquartile rangeby subtracting 109 from 98.And our interquartile range is 11.So it's called the interquartile range,because if you think of the range 0 to 100,there are four quarters, 0 to 25, 25 to 50, 50 to 75, and 75

  • 02:57

    DANIEL LITTLE [continued]: to 100.And we're dealing with only the range from 25% to 75%of our data.So the range describes what spread of scoresthat we actually observe.What are the spread of scores that we have?The remaining measures of variability that I will discussare all related to one another by virtueof how they are computed.

  • 03:19

    DANIEL LITTLE [continued]: The measures that I want to talk to youabout are called the sum of squares, the variance,and the standard deviation.To compute the sum of squares, what we would dois take each score and subtract from it the mean.So we would first need to work outwhat the mean is for this set of numbers, which is 102.56.

  • 03:43

    DANIEL LITTLE [continued]: Then what we do is we take each one of these numbers, whichI've indicated here as x subscript i.The i means that we'll iterate through each one of our numbershere, all the way down through all of the rest of the numbers.We will take each one of those numbers,subtract off the mean of 102.56, and square that value.

  • 04:04

    DANIEL LITTLE [continued]: Then to compute the sum of squares, what we would dois simply add up all of those squared deviations.This Greek letter here is called a capital Sigma,and it's used to indicate that wewant to apply the sum across all of our data values.If we do that, we end up with a sum

  • 04:26

    DANIEL LITTLE [continued]: of squares which indicates that the sum of squares is 2,114.4.This particular measure is important in the computationof ANOVA.Now, variance is very much related to the sum of squares.The variance is the expected value of the squared deviation

  • 04:48

    DANIEL LITTLE [continued]: from the mean.In other words, variance is the average squared deviationfrom the mean.So to compute the variance, what we would firstdo is compute the sum of squares,as I indicated on the previous slide,and then divide by the total number of data pointsthat we have, indicated here by n.So on the previous slide, if we just step back,

  • 05:11

    DANIEL LITTLE [continued]: we see that the sum of squares was 2,114.4.If we divide that by our total number of data points, whichis 18, then we end up with a variance of 117.47.

  • 05:31

    DANIEL LITTLE [continued]: Finally, our last measure of variability,which is also related to the sum of squares and the variance,is called the standard deviation.The standard deviation is simply the square rootof the variance.To compute the standard deviation, what we would dois compute the variance and take the square root.If we do that, we find that the standard deviationof this particular data set is 10.84,

  • 05:52

    DANIEL LITTLE [continued]: which is very similar to what we specifiedfor the normal distribution that the data were sampled from.Each of these measures of variabilitytell us something about the spread of the distribution,but they tell us very little about the location of our data.In order to understand the location of the data,we need to look at measures of central tendency, which

  • 06:13

    DANIEL LITTLE [continued]: I discuss in another tutorial video.

Video Info

Series Name: Statistics for Psychology

Episode: 17

Publisher: University of Melbourne

Publication Year: 2014

Video Type:Tutorial

Methods: Standard deviations, Normal distribution, Analysis of variance

Keywords: mathematical concepts

Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:

Keywords:

Abstract

In this final installment in his series on statistics for psychology, Professor Daniel Little offers a reference guide and glossary of concepts used throughout. He focuses on identifying and explaining the factors that define distribution variability.

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Measures of Variability: Range, Sum of Squares, Variance & Standard Deviation

In this final installment in his series on statistics for psychology, Professor Daniel Little offers a reference guide and glossary of concepts used throughout. He focuses on identifying and explaining the factors that define distribution variability.

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