Measures of central tendency mean, median and mode .ppt

Learn about Dispersion

1. Measures of Central TendencyMeasures of Central Tendency
oror
Measures of LocationMeasures of Location
oror
Measures of AveragesMeasures of Averages

2. INVESTIGATIONINVESTIGATION
Data Colllection
Data Presentation
Tabulation
Diagrams
Graphs
Descriptive Statistics
Measures of Location
Measures of Dispersion
Measures of Skewness &
Kurtosis
Inferential Statistiscs
Estimation Hypothesis
Testing
Ponit estimate
Inteval estimate
Univariate analysis
Multivariate analysis

3. 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.

4. 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.

5. 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.

6. Measure of central Measure of central
tendencytendency


N

N
1 i
ix


n
X
n
1 i
ix

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.

7. MeanMean
PopulationPopulation
SampleSample
N
X

n
X
X


“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

8. Mean: Example
Data: {1,3,6,7,2,3,5}
• number of observations: 7
•Sum of observations: 27
•Mean:
3.9

9. Simple Frequency Simple Frequency
DistributionsDistributions
namename XX
Student1Student1 2020
Student2Student2 2323
Student3Student3 1515
Student4Student4 2121
Student5Student5 1515
Student6Student6 2121
Student7Student7 1515
Student8Student8 2020
raw-score distribution frequency distribution
ffXX
331515
222020
222121
112323

f
N
Mean

10. 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.

11. Pros and Cons of the MeanPros and Cons of the Mean
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.

12. Some Important Properties of the Some Important Properties of the
MeanMean
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

13. 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+1
2

14. 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.

15. ModeMode
Most frequently occurring valueMost frequently occurring value
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

16. 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

17. Pros and Cons of the ModePros and Cons of the Mode
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.

18. 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
2
The mode is 34 years.
 

19. 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 mathematically
50% of sample is above the median, 50% of sample is
below the median.
 

20. 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?
 

21. Calculating the Mean from Calculating the Mean from
a Frequency Distributiona Frequency Distribution
# of Children(Y)
0
1
2
3
4
5
6
7
Total
Frequency(f)
12
25
733
333
183
26
15
12
1339
Frequency*Y(fY)
0
25
1466
999
732
130
90
84
3526
6.2
1339
3526


N
fY
Y

24. MEASURES OF MEASURES OF Central Central
TendencyTendency

Geometric MeanGeometric Mean
& Harmonic Mean& Harmonic Mean

26. The Shape of DistributionsThe Shape of Distributions
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.
?

27. SymmetricalSymmetrical
DistributionsDistributions
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.

28. 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
F
r
e
q
u
e
n
c
y
400
300
200
100
0
Std. Dev = 2.97
Mean = 13.4
N = 975.00

29. Skewed Distribution:Skewed Distribution:
Skewed DistributinSkewed Distributin
FFew extreme values on one side of the ew extreme values on one side of the
distribution or on the other.distribution 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)

30. Positively Skewed Positively Skewed
DistributionDistribution
GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE
4.03.02.01.0
GOVT INVESTIGATE WORKERS ILLEGAL DRUG USE
F
r
e
q
u
e
n
c
y
500
400
300
200
100
0
Std. Dev = .39
Mean = 1.1
N = 474.00
Mean=1.13
Median=1.0

31. Negatively Skewed Negatively Skewed
distributiondistribution
FAVOR PREFERENCE IN HIRING BLACKS
4.03.02.01.0
FAVOR PREFERENCE IN HIRING BLACKS
F
r
e
q
u
e
n
c
y
600
500
400
300
200
100
0
Std. Dev = .98
Mean = 3.3
N = 908.00
Mean=3.3
Median=4.0

32. Mean, Median and ModeMean, Median and Mode

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

35. 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

36. 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

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x
x
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