Central Tendency

1
Central Tendency
2
Numerical DataProperties Measures
Numerical Data
Properties
Central
RelativeStanding
Variation
Tendency
Mean
Range
Percentiles
Interquartile Range
Median
Zscores
Mode
Variance
Standard Deviation
3
Mean

• Measure of central tendency
• Most common measure
• Acts as balance point
• Affected by extreme values (outliers)
• Formula (sample mean)

4
Mean Example

• Raw Data 10.3 4.9 8.9 11.7 6.3 7.7

n
?
X
i
X
X
X
X
X
X
?
?
?
?
?
1
2
3
4
5
6
i
?
1
X
?
?
n
6
?
?
?
?
?
10
3
4
9
8
9
11
7
6
3
7
7
.
.
.
.
.
.
?
6
?
8
30
.
5
Numerical DataProperties Measures
Numerical Data
Properties
Central
RelativeStanding
Variation
Tendency
Mean
Range
Percentiles
Median
Interquartile Range
Zscores
Mode
Variance
Standard Deviation
6
Median

• Measure of central tendency
• Middle value in ordered sequence
• If n is odd, middle value of sequence
• If n is even, average of 2 middle values
• Position of median in sequence
• Not affected by extreme values

7
Median Example Odd-Sized Sample

• Raw Data 24.1 22.6 21.5 23.7 22.6
• Ordered 21.5 22.6 22.6 23.7 24.1
• Position 1 2 3 4 5

?
?
n
1
5
1
Positioning
Point
?
?
?
3
0
.
2
2
Median
?
22
6
.
8
Median Example Even-Sized Sample

• Raw Data 10.3 4.9 8.9 11.7 6.3 7.7
• Ordered 4.9 6.3 7.7 8.9 10.3 11.7
• Position 1 2 3 4 5 6

?
?
n
1
6
1
Positioning
Point
?
?
?
3
5
.
2
2
?
7
7
8
9
.
.
Median
?
?
8
30
.
2
9
Numerical DataProperties Measures
Numerical Data
Properties
Central
RelativeStanding
Variation
Tendency
Mean
Range
Percentiles
Interquartile Range
Median
Zscores
Mode
Variance
Standard Deviation
10
Mode

• Measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• May be no mode or several modes
• May be used for quantitative or qualitative data

11
Mode Example

• No ModeRaw Data 10.3 4.9 8.9 11.7 6.3 7.7
• One ModeRaw Data 6.3 4.9 8.9 6.3 4.9 4.9
• More Than 1 ModeRaw Data 21 28 28 41 43 43

12
Thinking Challenge

• Youre a financial analyst for Prudential-Bache
  Securities. You have collected the following
  closing stock prices of new stock issues 17,
  16, 21, 18, 13, 16, 12, 11.
• Describe the stock pricesin terms of central
  tendency.

13
Central Tendency Solution

• Mean

n
?
X
i
X
X
X
?
?
?

1
2
8
i
?
1
X
?
?
n
8
?
?
?
?
?
?
?
17
16
21
18
13
16
12
11
?
8
?
15
5
.
14
Central Tendency Solution

• Median
• Raw Data 17 16 21 18 13 16 12 11
• Ordered 11 12 13 16 16 17 18 21
• Position 1 2 3 4 5 6 7 8

?
?
n
1
8
1
Positioning Point
?
?
?
4
5
.
2
2
?
16
16
Median
?
?
16
2
15
Central Tendency Solution
Mode Raw Data 17 16 21 18 13 16 12 11 Mode
16
16
Summary of Central Tendency Measures
Measure
Formula
Description
Mean
Balance Point
??
X
/
n
i

Median
(
n
1)
Middle Value
Position
2
When Ordered
Mode
none
Most Frequent
17
Shape
18
Shape

• Describes how data are distributed
• Measures of Shape
• Skew Symmetry

Right-Skewed
Left-Skewed
Symmetric
Mean

Median
Mean


Median


Median

Mean
19
Variation
20
Numerical DataProperties Measures
Numerical Data
Properties
Central
RelativeStanding
Variation
Tendency
Range
Mean
Percentiles
Interquartile Range
Median
Zscores
Mode
Variance
Standard Deviation
21
Range

• Measure of dispersion
• Difference between largest smallest
  observations Range Xlargest Xsmallest
• Ignores how data are distributed

7
8
9
10
7
8
9
10
Range 10 7 3
Range 10 7 3
22
Numerical DataProperties Measures
Numerical Data
Properties
Central
RelativeStanding
Variation
Tendency
Mean
Range
Percentiles
Interquartile Range
Median
Zscores
Mode
Variance
Standard Deviation
23
Variance Standard Deviation

• Measures of dispersion
• Most common measures
• Consider how data are distributed

X
8.3
4
6
10
12
8
24
Sample Variance Formula

n - 1 in denominator! (Use N if Population
Variance)
25
Sample Standard Deviation Formula
2
S
S
?
n
(
)
2
?
X
X
?
i
i
?
1
?
n
?
1
(
)
(
)
(
)
2
2
2
X
X
X
X
X
X
?
?
?
?
?
?

n
1
2
?
n
?
1
26
Variance Example

• Raw Data 10.3 4.9 8.9 11.7 6.3 7.7

n
n
(
)
2
?
?
X
X
X
?
i
i
2
i
i
1
1
?
?
S
X
8
3
?
?
?
where
.
n
n
1
?
(
)
(
)
(
)
2
2
2
10
3
8
3
4
9
8
3
7
7
8
3
?
?
?
?
?
?
.
.
.
.
.
.

2
S
?
6
1
?
6
368
?
.
27
Thinking Challenge

• Youre a financial analyst for Prudential-Bache
  Securities. You have collected the following
  closing stock prices of new stock issues 17, 16,
  21, 18, 13, 16, 12, 11.
• What are the variance and standard deviation of
  the stock prices?

28
Variation Solution
Sample Variance Raw Data 17 16 21 18 13 16 12 11
n
n
(
)
2
?
?
X
X
X
?
i
i
2
i
i
1
1
?
?
S
X
15
5
?
?
?
where
.
n
n
1
?
(
)
(
)
(
)
2
2
2
17
15
5
16
15
5
11
15
5
?
?
?
?
?
?
.
.
.

2
S
?
8
1
?
11
14
?
.
29
Variation Solution

• Sample Standard Deviation

n
(
)
2
?
X
X
?
i
2
i
?
1
S
S
?
?
?
?
11
14
3
34
.
.
n
?
1
30
Summary of Variation Measures
Measure
Formula
Description
X

X
Total Spread
Range
largest
smallest
Dispersion about
Standard Deviation
Sample Mean
(Sample)
Dispersion about
Standard Deviation
Population Mean
(Population)
2
Squared Dispersion
Variance
?
(
X
?
X
)
?
i
about Sample Mean
(Sample)
n
1
31
Interpreting Standard Deviation
32
Interpreting Standard Deviation Chebyshevs
Theorem

• Applies to any shape data set

33
Interpreting Standard Deviation Chebyshevs
Theorem
34
Chebyshevs Theorem Example

• Previously we found the mean closing stock price
  of new stock issues is 15.5 and the standard
  deviation is 3.34.
• Use this information to form an interval that
  will contain at least 75 of the closing stock
  prices of new stock issues.

35
Chebyshevs Theorem Example

• At least 75 of the closing stock prices of new
  stock issues will lie within 2 standard
  deviations of the mean.
• x 15.5 s 3.34

(x 2s, x 2s) (15.5 23.34, 15.5
23.34) (8.82, 22.18)
36
Interpreting Standard Deviation Empirical Rule

• Applies to data sets that are mound shaped and
  symmetric
• Approximately 68 of the measurements lie in the
  interval µ s to µ s
• Approximately 95 of the measurements lie in the
  interval µ 2s to µ 2s
• Approximately 99.7 of the measurements lie in
  the interval µ 3s to µ 3s

37
Interpreting Standard Deviation Empirical Rule
µ 3s µ 2s µ s µ µ s
µ 2s µ 3s
38
Empirical Rule Example

• Previously we found the mean closing stock price
  of new stock issues is 15.5 and the standard
  deviation is 3.34. If we can assume the data is
  symmetric and mound shaped, calculate the
  percentage of the data that lie within the
  intervals x s, x 2s, x 3s.

39
Empirical Rule Example