5 Basic Descriptive Statistics Dispersion-1

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Numeracy & Quantitative Methods:Numeracy for Professional Purposes

Laura Lake


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Descriptive statistics conducting analysis on one variableat a time or univariate analysis.

Common approaches to univariate analysis:

Measures of distribution

Measures of central tendency

Measures of dispersion

Recap: univariate analysis


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Measures of dispersion: statistical measures that summarisethe amount of spread or variation in the distribution of valuesin a variable.

So, how values are spread within a distribution.

There are a number of different measures (applicable tointerval or ratio data):

RangeStandard deviation

Variance



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Type Description

RangeDifference between the highest (maximum)and lowest (minimum) value in the distributionof values

Variance The measure of the spread.

Standard deviation Shows the relation that a set of data has to themean of the sample data.


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Range is simply the difference between the highest andlowest value in the distribution of values.

Example:

Weekly income of 10 people:

Range is maximum income minus minimumincome: 330-180 = 150.

Range

180 220 280 320 280 180 350 280 330 220


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Of course, ordinal data can be ordered and so can giveinformation on range.

Example:

Survey question How useful did you find the book?

Range is from very useful to very un-useful.

Range using ordinal

data

Veryuseful

VeryUn-

usefulUseful

Un-useful

VeryUn-

usefulUseful

Veryuseful

Veryuseful

UsefulUn-

useful


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Inter quartile range (IQR) is another range measure but thistime looks at the data in terms of quarters or percentiles.

The range of data is divided into four equal percentiles or

quarters (25%).

Inter quartile range

Min Max

Q2

Median

50th Percentile

Q1

25th percentile

Q3

75th percentile

IQR

Range


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IQR is the range of the middle 50% of the data. Therefore,because it uses the middle 50%, it is not affected by outliers orextreme values.

Outliers variables that are the extreme lower or upper endof the distribution. They are atypical, infrequent observations.

These will influence the mean (arithmetic). Why?

10 people record their height: 160, 162, 164, 166, 168, 170, 172,174, 176 and 200 cm tall. With those values the mean is 171cm.200cm is the outliertake it out and the mean is 168cm.

Inter quartile range


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Where the mean is a measure of the centre of a group ofnumbers, the variance is the measure of the spread.

It involves measuring the distance between each of the

values and the mean.

To calculate the variance :

1. calculate the mean

2. for each value in the distribution subtract the meanand then square the result (the squared difference)

3. calculate the average of those squared differences.

Variance


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= Sum of (observed value mean score) 2

Total number of scores -1

The larger the variance value the further the observed values

of the data set are dispersed from the mean. A variance value of zero means all observed values are thesame as the mean.

Variance

1

2

2

N

XXs

i


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Standard deviation: how far on average each value is fromthe mean.

Problem with variance: because the differences are squared,

the units of variance are not the same as the units of the data.

This can make interpretation of the results problematic.

If the variance is square rooted, the units of variance then

correspond to those of the data set.

This square rooting of the variance is reported as thestandard deviation.

Standard deviation


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So, in most disciplines, standard deviation is used morefrequently than variance.

Chart example of standard deviation.

Standard deviation scores are used to generate standardised or zscores.

oStandardised scores are individual values expressed in units of

standard deviation from the mean.oUsed to compare variables with different unit measures.

Standard deviation


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Standard deviation = The square root of the variance.

As it is square rooted the results correspond to the originaldata units. E.g. if the variable is height recorded in cm thenthe standard deviation can be interpreted as cm.

Standard deviation

1

2

N

XXs


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Appropriate descriptive

statistics: summary

Level ofmeasurement

Univariate analysis

Nominal

Frequency table: count, %, valid %, cumulative %.

Measure of central tendency: modeMeasure of dispersion: no measure.

OrdinalFrequency table: count, %, valid %, cumulative %.Measure of central tendency: mode, medianMeasure of dispersion: no measure.

Interval/Ratio

Frequency table: count, %, valid %, cumulative %.Measure of central tendency: mode, median, meanMeasure of dispersion: range, variance, standarddeviation


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Bryman, A. (2008) Social Research Methods. 3rd Ed. Oxford:

Oxford University Press.

David, M. and Sutton, C. (2011) Social Research : An Introduction.

2nd ed. London: Sage.

References


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Author Laura Lake

Institute University of Plymouth

TitleNumeracy & Quantitative Methods

Numeracy for Professional Purposes

Description Basic Descriptive Statistics: Introduction

Date Created May 2011

Educational Level Level 4

Keywords

Learning from WOeRK Work Based Learning WBL Continuous

Professional Development CPD Research UKOER LFWOER Measures of

dispersion, range, variance, standard deviation.

Back page originally developed by the OER phase 1 C-Change project

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5 Basic Descriptive Statistics Dispersion-1