Measures of Central Tendency or Measures of Location or Measures of Averages

Measures of Central TendencyMeasures of Central Tendency

oror

Measures of LocationMeasures of Location

oror

Measures of AveragesMeasures of Averages

Descriptive StatisticsDescriptive Statistics

The goal of descriptive statistics is to The goal of descriptive statistics is to summarize a collection of data in a clear and summarize a collection of data in a clear and understandable way.understandable way. What is the pattern of scores over the range of What is the pattern of scores over the range of

possible values?possible values? Where, on the scale of possible scores, is a point Where, on the scale of possible scores, is a point

that best represents the set of scores?that best represents the set of scores? Do the scores cluster about their central point or do Do the scores cluster about their central point or do

they spread out around it?they spread out around it?

Central TendencyCentral Tendency

Measure of Central Tendency:Measure of Central Tendency: A single summary score that best describes the A single summary score that best describes the

central location of an entire distribution of scores.central location of an entire distribution of scores. The typical score.The typical score. The center of the distribution.The center of the distribution.

One distribution can have multiple locations where One distribution can have multiple locations where scores cluster.scores cluster. Must decide which measure is best for a given situation.Must decide which measure is best for a given situation.

Central TendencyCentral Tendency

Measures of Central Tendency:Measures of Central Tendency: MeanMean

The sum of all scores divided by the number of The sum of all scores divided by the number of scores.scores.

MedianMedian The value that divides the distribution in half The value that divides the distribution in half

when observations are ordered.when observations are ordered. ModeMode

The most frequent score.The most frequent score.

Measure of central Measure of central tendencytendency

N

N

1 iix

n X

n

1 iix

Population Sample

Arithmetic Mean (Mean)

Definition:Sum of all the observation s divided by the number of the observations

The arithmetic mean is the most common measure of the central location of a sample.

MeanMean

PopulationPopulation

SampleSample

N

X

n

XX

“mu”

“X bar”

“sigma”, the sum of X, add up all scores

“n”, the total number of scores in a sample

“N”, the total number of scores in a population

“sigma”, the sum of X, add up all scores

Mean: Example

Data: {1,3,6,7,2,3,5}

• number of observations: 7•Sum of observations: 27•Mean: 3.9

Simple Frequency Simple Frequency DistributionsDistributions

namename XX

Student1Student1 2020








raw-score distribution frequency distribution

ff XX

33 1515

22 2020

22 2121

11 2323

f

NMean

MeanMean

Is the balance point of a distribution.Is the balance point of a distribution. The sum of negative deviations from the mean The sum of negative deviations from the mean

exactly equals the sum of positive deviations exactly equals the sum of positive deviations from the mean.from the mean.

Pros and Cons of the Pros and Cons of the MeanMean

ProsPros Mathematical center of a Mathematical center of a

distribution.distribution. Good for interval and ratio Good for interval and ratio

data.data. Does not ignore any Does not ignore any

information.information. Inferential statistics is Inferential statistics is

based on mathematical based on mathematical properties of the mean.properties of the mean.

ConsCons Influenced by extreme Influenced by extreme

scores and skewed scores and skewed distributions.distributions.

May not exist in the data.May not exist in the data.

Some Important Properties of the MeanSome Important Properties of the Mean

Interval-Ratio Level of MeasurementInterval-Ratio Level of Measurement

Center of Gravity(the mean balances Center of Gravity(the mean balances

all the scores). all the scores).

Sensitivity to ExtremesSensitivity to Extremes

Median Definition: The value that is larger than half the population and smaller than half the population n is odd: the median score 5, 8, 9, 10, 28 median = 9

n is even: the th score

6, 17, 19, 20, 21, 27 median = 19.5

n+12

Pros and Cons of MedianPros and Cons of Median

ProsPros Not influenced by Not influenced by

extreme scores or extreme scores or skewed distributions.skewed distributions.

Good with ordinal Good with ordinal data.data.

Easier to compute Easier to compute than the mean.than the mean.

ConsCons May not exist in the May not exist in the

data.data. Doesn’t take actual Doesn’t take actual

values into account.values into account.

ModeModeMost frequently occurring Most frequently occurring valuevalue

Data {1,3,7,3,2,3,6,7}• Mode : 3

Data {1,3,7,3,2,3,6,7,1,1}• Mode : 1,3

Data {1,3,7,0,2,-3, 6,5,-1}• Mode : none

Central Tendency Central Tendency Example: ModeExample: Mode

52, 76, 100, 136, 186, 196, 205, 150, 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891750, 791, 891

Mode: most frequent observationMode: most frequent observation Mode(s) for hotel rates:Mode(s) for hotel rates:

264, 317, 384264, 317, 384

Pros and Cons of the Pros and Cons of the ModeMode

ProsPros Good for nominal Good for nominal

data.data. Easiest to compute Easiest to compute

and understand.and understand. The score comes The score comes

from the data set.from the data set.

ConsCons Ignores most of the Ignores most of the

information in a information in a distribution.distribution.

Small samples may Small samples may not have a mode.not have a mode.

Example: Central LocationExample: Central Location

Suppose the age in years of the first 10 subjects enrolled in your study are:

34, 24, 56, 52, 21, 44, 64, 34, 42, 46

Then the mean age of this group is 41.7 years

To find the median, first order the data:21, 24, 34, 34, 42, 44, 46, 52, 56, 64

The median is 42 +44 = 43 years

2The mode is 34 years.

Comparison of Mean and Median

• Mean is sensitive to a few very large (or small) values “outliers” so sometime mean does not reflect the quantity desired.

• Median is “resistant” to outliers

• Mean is attractive mathematically50% of sample is above the median, 50% of sample is below the median.

Suppose the next patient enrolls and their age is 97 years.How does the mean and median change?

To get the median, order the data:21, 24, 34, 34, 42, 44, 46, 52, 56, 64, 97

If the age were recorded incorrectly as 977 instead of 97, what would the new median be? What would the new mean be?

Calculating the Mean Calculating the Mean from a Frequency from a Frequency DistributionDistribution

# of Children(Y)01234567

Total

Frequency(f)1225

733333183261512

1339

Frequency*Y(fY)0

2514669997321309084

3526

6.21339

3526

N

fYY

MEASURES OF MEASURES OF Central Central TendencyTendency

Geometric MeanGeometric Mean

& Harmonic Mean& Harmonic Mean

The Shape of The Shape of DistributionsDistributions

Distributions can be either Distributions can be either symmetricalsymmetrical or or skewedskewed, depending on , depending on whether there are more frequencies at whether there are more frequencies at one end of the distribution than the one end of the distribution than the other.other.

?

SymmetricalSymmetricalDistributionsDistributions

A distribution is symmetrical if the A distribution is symmetrical if the frequencies at the right and left tails of frequencies at the right and left tails of the distribution are identical, so that if the distribution are identical, so that if it is divided into two halves, each will it is divided into two halves, each will be the mirror image of the other. be the mirror image of the other.

In a symmetrical distribution the In a symmetrical distribution the mean, median, and mode are identical.mean, median, and mode are identical.

Almost Symmetrical distributionAlmost Symmetrical distribution

Mean=13.4

Mode=13.0

HIGHEST YEAR OF SCHOOL COMPLETED

20.017.515.012.510.07.55.02.50.0

HIGHEST YEAR OF SCHOOL COMPLETED

Fre

qu

en

cy

400

300

200

100

0

Std. Dev = 2.97

Mean = 13.4

N = 975.00

Skewed Distribution:Skewed Distribution:

Skewed DistributinSkewed DistributinFFew extreme values on one side of the distribution ew extreme values on one side of the distribution or on the other.or on the other. Positively skewedPositively skewed distributions: distributions:

distributions which have few distributions which have few extremely high values extremely high values (Mean>Median)(Mean>Median)

Negatively skewed distributions: Negatively skewed distributions:

distributions which have few distributions which have few

extremely low values(Mean<Median)extremely low values(Mean<Median)

Positively Skewed Positively Skewed DistributionDistribution

GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE

4.03.02.01.0

GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE

Fre

qu

en

cy

500

400

300

200

100

0

Std. Dev = .39

Mean = 1.1

N = 474.00

Mean=1.13

Median=1.0

Negatively Skewed Negatively Skewed distributiondistribution

FAVOR PREFERENCE IN HIRING BLACKS

4.03.02.01.0

FAVOR PREFERENCE IN HIRING BLACKS

Fre

qu

en

cy

600

500

400

300

200

100

0

Std. Dev = .98

Mean = 3.3

N = 908.00

Mean=3.3

Median=4.0

Mean, Median and Mean, Median and ModeMode

DistributionsDistributions

Bell-Shaped (also Bell-Shaped (also known as symmetric” or known as symmetric” or “normal”)“normal”)

Skewed:Skewed: positively (skewed to the positively (skewed to the

right) – it tails off toward right) – it tails off toward larger valueslarger values

negatively (skewed to negatively (skewed to the left) – it tails off the left) – it tails off toward smaller values toward smaller values

Choosing a Measure of Central Choosing a Measure of Central TendencyTendency

IF variable is Nominal..IF variable is Nominal.. ModeMode IF variable is Ordinal...IF variable is Ordinal... Mode or Median(or both)Mode or Median(or both) IF variable is Interval-Ratio and distribution is IF variable is Interval-Ratio and distribution is

Symmetrical…Symmetrical… Mode, Median or Mean Mode, Median or Mean IF variable is Interval-Ratio and distribution is IF variable is Interval-Ratio and distribution is

Skewed…Skewed… Mode or MedianMode or Median

EXAMPLE:EXAMPLE:

(1) 7,8,9,10,11 n=5, x=45, =45/5=9(1) 7,8,9,10,11 n=5, x=45, =45/5=9

(2) 3,4,9,12,15 n=5, x=45, =45/5=9(2) 3,4,9,12,15 n=5, x=45, =45/5=9

(3) 1,5,9,13,17 n=5, x=45, =45/5=9(3) 1,5,9,13,17 n=5, x=45, =45/5=9

S.D. : (1) 1.58 (2) 4.74 (3) 6.32 S.D. : (1) 1.58 (2) 4.74 (3) 6.32

x

x

x

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Measures of Central Tendency or Measures of Location or Measures of Averages