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IMS 502IMS 502

Information Analysis for Decision MakingInformation Analysis for Decision Making

DescriptiveDescriptive StatisticsStatistics

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Content Measure of Location

Measure of Spread

Measure of Shape

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Learning Outcomes After completing this chapter, you should be

able to:Able to conduct appropriate measure of location,

measure of spread & measure of shape

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Concept of Descriptive

Statistics

Describing Data NumericallyMeasure of Location

(Central Tendency)

Measure of Spread

(Variability)

Measure of Shape

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StatisticsMeasure of Location

(Central Tendency)

MeanMean

MedianMedian

ModeMode

Descriptive information about the single numerical value that isconsidered to be the most typicalof the values of a quantitative variable.

A measure of central tendency is a measure which indicates where the

middlemiddle of the data is. Three common measures of central tendency are:

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Measures of Central

Tendency

Statistics to represent the centre of a

distributionMode (most frequent)

Median (50th percentile)

Mean (average)

Choice of measure dependent on

Type of data

Shape of distribution (esp. skewness)

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Measures of Central

Tendency

Measure of Central Tendency

Level of

Measurement

Mode Median Mean

Nominal X

Ordinal X X X

Interval X X X

Ratio X? X X

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Measures of Central

Tendency Choosing a measure of central tendency

Use the Mode When : 1. The variable is measured at the nominal level.

2. You want a quick and easy measure for ordinal and

interval-ratio variables.

3. You want to report the most common score.

Use the Median When : 1. The variable is measured at the ordinal level.2. A variables measured at the interval-ratio level has a

highly skewed distribution.

3. You want to report the central score. The median

always lies at the exact center of a distribution.

Use the Mean When : 1. The variable is measured at the interval-ratio level

(except when the variable is highly skewed).

2. You want to report the typical score. The mean is the

fulcrum that exactly balances all of the scores.

3. You anticipate additional statistical analysis.

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Measures of Central

Tendency Mean

Is the average of the data

The Population Mean: = which is usually unknown, then we use the

sample mean to estimate or approximate it.

The Sample Mean:=

x

1

N

ii

N

X=

1

n

i

i

n

x=

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Measures of Central

Tendency

Example:Example:Example:Example:Here is a random sample of size 10 of ages, where

1 = 42, 2 = 28, 3 = 28, 4 = 61, 5 = 31, 6 = 23, 7 = 50, 8 = 34, 9 = 32, 10 = 37.

= (42 + 28 + + 37) / 10 = 36.6x

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Measures of Central

Tendency Properties of the Mean

UniquenessUniqueness: For a given set of data there isone and only one mean.

SimplicitySimplicity: It is easy to understand and to

compute.

Affected by extreme valuesAffected by extreme values: Since all valuesenter into the computation.

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Measures of Central

Tendency

Example:Example:Example:Example:Assume the values are 115, 110, 119, 117, 121 and 126.

The mean = 118.

But assume that the values are 75, 75, 80, 80 and 280.

The mean = 118, a value that is not representativenot representative of the set of data asa whole.

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Statistics Median

When orderingordering the data, it is the observation thatdivide the set of observations into two equal partsinto two equal parts

such that half of the data are before it and the other

are after it.

The median is the center point in a set of numbers

(50% above, 50% below)

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Statistics Median

Check to see which of the following tworules applies:

Rule One

Ifnn is odd, the median will be the middle ofobservations.

It will be the (n+1)/2 th ordered observation.

When n = 11, then the median is the 6th observation.

Example:Example:Example:Example:

Three is the median for the numbers 1, 1, 33, 4, 9

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Statistics Median

Check to see which of the following tworules applies:

Rule Two

Ifnn is even, there are two middle observations. The medianwill be the mean of these two middle observations.

It will be the (n+1)/2 th ordered observation.

When n = 12, then the median is the 6.5th observation,

which is an observation halfway between the 6th and 7thordered observation.

Example:Example:Example:Example:

23, 28, 28, 31, 32, 34, 37, 42, 50, 61.

Since n = 10, then the median is the 5.5th observation, i.e. =(32+34)/2 = 33

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Measures of Central

Tendency Properties of the Median

UniquenessUniqueness: For a given set of data there isone and only one median.

SimplicitySimplicity: It is easy to calculate.

It is not affected by extreme valuesIt is not affected by extreme values as is the

mean.

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Statistics Mode

The mode is simply the most frequentlyfrequentlyoccurring number.

If all values are different there is no modeno mode.

Sometimes, there are more than one modemore than one mode.

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Measures of Central

Tendency

Example:Example:Example:Example:Assume the values are 23, 28, 28, 31, 32, 34, 37, 42, 50, 61.

The median = 28 (repeated two times).

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Measures of Central

Tendency Properties of the Mode

Sometimes, it is not unique.not unique.

It may be used for describing qualitativedescribing qualitative

data.data.

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Measures of Central

Tendency The only procedure in SPSS that will produce all three

commonly used measure of location is Frequency.

To begin:

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Working Example(Pg. 59) One hundred tennis players participated

in a serving competition. Gender andnumber of aces were recorded for each

player. The data can be found in

Work4.sav on the iLearn web site that

accompanies this title.

Follow steps 1, 2, 3, 4, 6, 8 & 11.

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Exercises Use the Frequencies command to get all

three (3) measures of location. Write a sentence or two reporting each

measure. You may choose three(3)

variables to report on.

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Statistics

Measure of Spread

(Variability/Dispersion)

Give information on the spread or variability of the data values. Measures of deviation from the central tendency.

Non-parametric/non-normal: range, percentiles, min, max

Parametric: standard deviation (SD) & properties of the normal distribution

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Statistics

RangeRange

QuartilesQuartiles

VarianceVarianceStandardStandard

DeviationDeviation

Measure of Spread

(Variability/Dispersion)

Four common measures of spread are:

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Measures of Spread

Measure of Spread

Level of

Measurement

Range, Min/Max Percentile Standard Deviation

(SD)

Nominal

Ordinal X

Interval X X X?

Ratio X X X

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Measures of Spread Range (R)

The range is the difference between thehighest and lowest scores in a distribution.

If we examine the marks of the 100 students

above, then we can see that the highestscore was 85 and the lowest was 35.

Therefore, the range is 50 (85 35 = 50).

The range is limited as a means of tellingabout the general spread of a group of data,

it does set the boundaries of the scores.

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Measures of Spread Quartiles (Q)

The quartiles split the ordered data into fourquarters:

Q1, or 25th percentile -- 25.0% of theobservations

Q2, is the median -- 50.0% of the observations Q3, or 75th percentile -- 75.0% of the

observations

The difference between Q3 - Q1 is called theinter-quartile range, or IQR.

25%25% 25%25% 25%25% 25%25%

QQ11 QQ22 QQ33

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Measures of Spread Variance

The variance is the square of the standarddeviation.

The lower the variance, the more accurately

the mean represents the scores of all casesin a distribution of data.

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Measures of Spread Standard Deviation

The standard deviation provides theresearcher with an indicator of how scoresfor variables are spread around the mean

average.The higher the standard deviation, the morescores around the mean are spread out.

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Measures of Spread Using SPSS for measure of spread.

To begin:

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in a serving competition. Gender andnumber of aces were recorded for eachplayer. The data can be found inWork4.sav on the iLearn web site that

accompanies this title. Follow steps 1, 2, 3, 4, 5, 7, 8 & 11.

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Exercises Choose three(3) variables to work on.

Write a few sentences summarizing thesetables for each under measurement ofspread.

Describe the difference (if any).

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Statistics

SkewnessKurtosis

Measure of ShapeMeasure of Shape

To describes how data are distributed. Use the normal curve (combination of mean & standard deviation) to construct

precise descriptive statements. Two common measures of shape are:

Normal

Negative Skewness

Positive Skewness

Platykurtic

Leptokurtic

is a measure of symmetry, or moreprecisely, the lack of symmetry.

is a measure of whether the data are

peaked or flat relative to a normaldistribution.

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Measure of Shape - Skewness

Mean = Median = ModeMean = Median = Mode

Coefficient = 0Coefficient = 0

Mean < Median < ModeMean < Median < Mode

Coefficient = NegativeCoefficient = Negative

Mode < Median < MeanMode < Median < Mean

Coefficient = PositiveCoefficient = Positive

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Measure of Shape - Kurtosis

Data distribution with largestandard deviation. Data sets with low kurtosis tendto have a flat top near the mean

rather than a sharp peak. The data will be far away fromthe mean.

Kurtosis, 2 < 0 (Platykurtic)

Data distribution with small

standard deviation. Data sets with high kurtosis tend

to have a distinct peak near the

mean, decline rather rapidly, andhave heavy tails.

The data will cluster around orclose to the Mean.

Kurtosis, 2 > 0 (Leptokurtic)

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Measure of Shape Using SPSS for measure of shape.

To begin:

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in a serving competition. Gender andnumber of aces were recorded for eachplayer. The data can be found inWork4.sav on the iLearn web site that

accompanies this title. Follow steps 1, 2, 3, 4, Click on Skewness

& Kurtosis, 8, 9, 10 & 11.

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Exercise Choose three(3) variables to work on.

Write a few sentences summarizing thesetables for each under measurement ofshape.

Describe the difference (if any).

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IMS 504-Week 4&5 new