This document discusses measures of central tendency and variation for numerical data. It defines and provides formulas for the mean, median, mode, range, variance, standard deviation, and coefficient of variation. Quartiles and interquartile range are introduced as measures of spread less influenced by outliers. The relationship between these measures and the shape of a distribution are also covered at a high level.

1. MEASURES OF LOCATION and
VARIATION for 1 variable

2. Lectures 3+4+5 Topics
•Measures of Central Tendency for
numerical and categorical data
Mean, Median, Mode + other means, Fractiles
•Measures of Variation for numerical and
binary data
The Range, Variance and
Standard Deviation
•Shape
Symmetric, Skewed, Skewness, Kurtosis

3. Summary Measures
Summary Measures
Central Tendency part
of Location
Mean
Median
Mode
Variation
Range
Variance
Coefficient of
Variation
Standard Deviation
Fractiles

4. Measures of Central Tendency
Central Tendency
Mean Median Mode
n
x
n
i
i å=
1

5. The Mean (Arithmetic mean,
Average)
•It is the Arithmetic Average of data values:
n
x
n
i 1
i å=
= xi + x2 + · · · +xn
n
•The Most Common Measure of Central Tendency
•Affected by Extreme Values (Outliers)
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
x =
Sample Mean

6. THE ARITHMETIC
MEAN
 This is the most popular and useful measure of central location
Sum of the observations
Mean =N umber of observations

7. THE ARITHMETIC
MEAN
Sample mean Population mean
ån
n
=
i=1
x xi
n
N
xi
N
åi=1
m =
Sample size Population size

8. The arithmetic
mean
• Example 1
The reported time spent on the Internet of 10 adults are 0, 7, 12, 5,
33, 14, 8, 0, 9, 22 hours. Find the mean time spent on the Internet.
=
+ + +
=
å
= =
...
10
10
1 2 10
10
x i 1 xi x x x
00 77 2222
1111..00 hhoouurrss
• Example 2
Suppose the telephone bills represent
the population of measurements ( 200). The population mean is
=
200
i 1 xi x 4422..1 1199 + x 3388..2 4455 + ... +
x
4455..200
7777
m = å = =
200
200
4433.5.599
THE ARITHMETIC
MEAN

9. WEIGHTED MEAN FOR DATA
GROUPED BY CATEGORIES OR
VARIANTS
i i
i
k
i
f
x = å x f
=1
å

10. When many of the measurements have the same value, the
measurement can be summarized in a frequency table. Suppose
the number of children in a sample of 16 families were recorded
as follows:
NUMBER OF CHILDREN 0 1 2 3
NUMBER OF FAMILIES 3 4 7 2
16 families
3(0) 4(1) 7(2) 2(3)
16 1.5
. ...
16
16
1 1 2 2 16 16
16
= å =1 = + + = + + + = x i xi fi x f x f x f

11. MEAN
 FOR TABULATED DATA BY CLASSES

12. APPROXIMATING DESCRIPTIVE
MEASURES FOR GROUPED DATA BY
CLASSES
 Approximating descriptive measures for grouped data may be
needed in two cases:
 when approximated values.suffices the needs,
 when only secondary grouped data are available.
x x f
= å x midpoint
i
1
=
k
i
i i
k
i
f
1
=
å
f frequency

13.  Example 3
 Approximate the mean (calculate the mean) of the telephone call
durations problem as represented by the frequency distribution
Class Class Frequency Midpoint
i limits fi xi xi fi
1 2-5 3 3.5 10.5
2 5-8 6 6.5 39.0
3 8-11 8 9.5 76.0
…. …. … …. …. .
6 17-20 2 18.5 37.0
Class Class Frequency Midpoint
i limits fi xi xi fi
1 2-5 3 3.5 10.5
2 5-8 6 6.5 39.0
3 8-11 8 9.5 76.0
…. …. … …. …. .
6 17-20 2 18.5 37.0
n =sample size= 30=f1+…+fn 312.0
n =sample size= 30=f1+…+fn 312.0
Real value :
x =
10.26
8 11 14 17 20 More
6.5

14. The Median
•Important Measure of Central Tendency
•In an ordered array, the median is the
“middle” number.
•If n is odd, the median is the middle number.
•If n is even, the median is the average of the 2
middle numbers.
•Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5

15. THE MEDIAN
 The Median of a set of observations is the value that
falls in the middle when the observations are arranged
in order of magnitude or ranked increasingly
Example
Find the median of the time spent on the internet
for the adults of example 1
Comment
Suppose only 9 adults were sampled
(exclude, say, the longest time (33))
Odd number of observations
Even number of observations
8.5
, 8
0, 00,, 50,, 75,, 87,, 8 , 9 , 192, ,1 124, ,1 242, ,2 323, 33 0, 0, 5, 7, 9, 12, 14, 22

16. MEDIAN
 Data Tabulated discretely – as ungrouped
 Data Tabulated by classes - estimation

17. MEDIAN AND MODE
 Median
1
å å=
n
+
i ( 1) - n
2
Me
Me-1
i 1
i
= +
x K
0 n
Me

18. The Mode
•A Measure of Central Tendency
•Value that Occurs Most Often
•Not Affected by Extreme Values
•There May Not be a Mode
•There May be Several Modes
•Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode

19. THE MODE
 The Mode of a set of observations is the variable
value that occurs most frequently.
 Set of data may have one mode (or modal class), or
two or more modes.
The modal class
For large data sets
the modal class is
much more relevant
than a single-value
mode.

20. MEDIAN AND MODE
 Mode
1
Mo = + D
1 2
0 x K
D + D

21. RELATIONSHIP AMONG MEAN,
MEDIAN, AND MODE
 If a distribution is symmetrical, the mean, median and mode
coincide
• If a distribution iiss nnoonn ssyymmmmeettrriiccaall,, aanndd
sskkeewweedd ttoo tthhee lleefftt oorr ttoo tthhee rriigghhtt,, tthhee
tthhrreeee mmeeaassuurreess ddiiffffeerr..
A positively skewed distribution
(“skewed to the right”)
A negatively skewed distribution
(“skewed to the left”)
Mode Mean
MeaMedian
n Median
Mode

22. OTHER MEANS
 Harmonic
 Geometric
 Square

23. FRACTILES
 Quartiles: 3
 Percentiles: 99

24. Summary Measures
Central Tendency
Mean
Median
Mode
n
x
n
i
i å=
1
Summary Measures
( )
n 1
s 2 = å x i
-
x
-
Variation
Range
Variance
2
Coefficient of
Variation
Standard Deviation

25. Measures of Variation
Variation
Variance Standard Deviation Coefficient of
Variation Population
Variance
Sample
Variance
Population
Standard
Deviation
Sample
Standard
Deviation
Range
CV S
ö çè
100% × ÷ø
=æ
X

26. The Range
• Measure of Variation
• Difference Between Largest & Smallest
Observations:
Absolute Range =
• Relative Range =
•Ignores How Data Are Distributed:
xLargest - xSmallest
7 8 9 10 11
12Range = 12 - 7 = 5
x x mean La Smallest ( ) / rgest -
7 8 9 10 11
12Range = 12 - 7 = 5

27. INTERQUARTILE RANGE
 Can eliminate some outlier problems by using the interquartile
range
 Eliminate high- and low-valued observations and calculate the
range of the middle 50% of the data
 Interquartile range = 3rd quartile – 1st quartile
IQR = Q3 – Q1

28. INTERQUARTILE RANGE
Median
(Q2)
Example:
X Xmaximum minimum Q1 Q3
25% 25% 25% 25%
12 30 45 57 70
Interquartile range
= 57 – 30 = 27

29. QUARTILES
Quartiles split the ranked data into 4 segments
with an equal number of values per segment
25% 25% 25% 25%
QQ
11
QQ
22
QQ
33
• TThhee ffiirrsstt qquuaarrttiillee,, QQ11,, iiss tthhee vvaalluuee ffoorr wwhhiicchh 2255%%
ooff tthhee oobbsseerrvvaattiioonnss aarree ssmmaalllleerr aanndd 7755%% aarree
llaarrggeerr
• QQ22 iiss tthhee ssaammee aass tthhee mmeeddiiaann ((5500%% aarree
ssmmaalllleerr,, 5500%% aarree llaarrggeerr))
• OOnnllyy 2255%% ooff tthhee oobbsseerrvvaattiioonnss aarree ggrreeaatteerr tthhaann
tthhee tthhiirrdd qquuaarrttiillee

30. QUARTILE FORMULAS
Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position: Q1 = 0.25(n+1)
Second quartile position: Q2 = 0.50(n+1)
(the median position)
Third quartile position: Q3 = 0.75(n+1)
where n is the number of observed values

31. • EExxaammppllee:: FFiinndd tthhee ffiirrsstt
qquuaarrttiillee
((nn == 99))
QQ11 == iiss iinn tthhee 00..2255((99++11)) == 22..55 ppoossiittiioonn ooff tthhee
rraannkkeedd ddaattaa
ssoo uussee tthhee vvaalluuee hhaallff wwaayy bbeettwweeeenn tthhee 22nndd aanndd 33rrdd
vvaalluueess,,
ssoo QQ11 == 1122..55
QUARTILES
Sample Ranked Data: 11 12 13 16 16 17 18 21 22

32. DEVIATION
 Individual deviation from the mean =
Overall deviation = 0, because
 Summing squared deviations
or
absolute values of the deviations
x mean i -
å(X - X ) = 0 i
å( - )2 X X i
| x x | å i -

33. Variance
•Important Measure of Variation
•Shows Variation About the Mean
• Computed as an arithmetic mean of
squared deviations or as a square mean of
individual deviations
•For the Population:
•For the Sample:
( )
N
= å Xi -
2
s 2 m
( )
s = å Xi -
X
1
2
2
-
n
For the Population: use N in the
denominator.
For the Sample : use n - 1
in the denominator.

34. Standard Deviation
•Most Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
( )
N
m 2 s
= å Xi -
( )
s = å Xi -
X
1
2
-
n
For the Population: use N in the
denominator.
For the Sample : use n - 1
in the denominator.

35. Sample Standard Deviation
( )
Xi X
= å -
1
2
-
n
s
Xi :
Data: 10 12 14 15 17 18 18
24
s =
n = 8 Mean =16
10 16 2 12 16 2 14 16 2 15 16 2 17 16 2 18 16 2 24 16 2
( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - )
8 -
1
= 4.2426

36. Comparing Standard Deviations
Data : X i : 10 12 14 15 17 18 18 24
N= 8 Mean =16
( )
s = å Xi -
X
1
2
-
n
= 4.2426
( )
N
s = å Xi -
m 2 = 3.9686
Value for the Standard Deviation is larger for data considered as a Sample.

37. Comparing Standard Deviations
Mean = 15.5
Data A - AGE
11 12 13 14 15 16 17 18 19 20 21 s = 3.338
Data B - AGE
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Data C - AGE

38. COEFFICIENT OF VARIATION
Measure of Relative Variation
Always a % or coefficient
Shows Variation Relative to Mean
Used to Compare 2 or More Groups
Formula ( for Sample):
CV S
ö çè
100% × ÷ø
= æ
X

39. COMPARING COEFFICIENT OF VARIATION
 Stock A: Average Price last year = $50
 Standard Deviation (sd) = $5
 Stock B: Average Price last year = $100
 (sd) = $5
CV S
ö çè
100% × ÷ø
= æ
X
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
Both average prices are
representatives

40. SHAPE
 Describes How Data Are Distributed between smallest and largest
values
 Measures of Shape:
 Symmetric or skewed
Right-Skewed or
Positively Skewed
Left-Skewed or
Positive Skew-ness Symmetric
Mean Median Mode Mean = Median = Mode Mode Median Mean

41. BOX PLOT – GRAPHICAL
PRESENTATION OF CTM

44. CENTRAL TENDENCY MEASURES
SUMMARY FOR 1 VARIABLE
 Discussed Measures of Central Tendency
 Mean, Median, Mode Addressed Measures of Variation  The Range, Variance,
 Standard Deviation, Coefficient of Variation
 Determined Shape of Distributions
 Symmetric or Skewed
Coefficient of skewness
Mean Median Mode Mean = Median = Mode Mode Median Mean