Descriptive Statistics
Measure of Central Tendency
Numerical Summary
Measures
Mean: the average value of the data.
If n observations are denoted by x1, x2, ..., xn,
their (sample) mean is
Describe Distribution with Numbers
• Measure of Center
• Measure of Variation
• Measure of Position
n
x + x + ... + xn
x= 1 2
=
n
∑x
i =1
i
n
1
Example: Birth weights (in lb) of 5 babies born
from a group of women under certain diet.
2
Example: (number of hysterectomies
performed by 15 male doctors)
7, 6, 8, 7, 7
27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20,
36, 59, 34, 28
⇒ mean = 41.33
Sol:
2 | 05578
3 | 13467
4|4
5 | 09
6|
7|
8 | 56
mean = (7 + 6 + 8 +7 + 7) / 5 = 35/5 = 7
→
[near the center of the data set]
3
Weighted Mean
Weighted Mean
Example: (Grade point average)
A student received 3 A’s, 5 B’s, 2 C’s.
Class (grade point, x)
4
3
Frequency (weight, w)
3
5
2
2
Average grade point =
=
4
3x4+5x3+2x2
3+5+2
weight
31
= 3.1
10
Weighted mean =
w1. x1 + w2. x2 + … + wk. xk
w1 + w2 + … + w k
=
Σw.x
Σw
Where w1, w2, … are the weights and x1, x2, …
are the values (or class midpoint or class mark).
5
6
Descriptive Stat - 1
Descriptive Statistics
Grouped Mean
Mean Estimation
Class
Frequency Class Mark
(w)
(x)
Median: of a data set is
w .x
90 - < 110
1
100
1x100
110 - < 130
2
120
2x120
130 - <150
3
140
3x140
150 - < 170
1
160
1x160
Total
7
Estimated mean =
„
„
the data value exactly in the middle of its
ordered list if the number of pieces of data is
odd,
the mean of the two middle data values in its
ordered list if the number of pieces of data is
even.
920
920
7
[median is not influenced by outliers and is best
for nonsymmetric distribution]
= 131.43
7
Example: (number of hysterectomies
performed by 15 doctors)
27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20,
36, 59, 34, 28
8
Example: (Birth weights for 6 infants.)
5, 7, 6, 8, 5, 9
ordered list => 5, 5, 6, 7, 8, 9
ordered list => 20, 25, 25, 27, 28, 31, 33,
34, 36, 37, 44, 50, 59, 85, 86
median = 34
median = (6+7) / 2 = 6.5
9
10
Example 1: (number of times visited
class website by 15 students)
27, 50, 33, 25, 86, 25, 85, 31, 37, 44,
20, 36, 59, 34, 28
Mode: of a data set is the observation
that occurs most frequently.
ordered list => 20, 25, 25, 27, 28, 31,
33, 34, 36, 37, 44, 50, 59, 85, 86
Mode = 25
Example 2: (Blood type of 15 students)
A, B, A, A, O, AB, A, A, B, B, O, O, A, A, A
Mode = A
11
A–8
B–3
O–3
AB – 1
12
Descriptive Stat - 2
Descriptive Statistics
Mean ?
What is a Modal class?
Skewed to the Right
Median ?
Class
90< - 110
110< - 130
130< - 150
150< - 170
170< - 190
190< - 210
210< - 230
230< - 250
250< - 270
270< - 290
Total
Frequency
2
2
4
2
7
3
1
0
0
1
22
Relative Freq.
2/22 = .091
2/22 = .091
4/22 = .182
2/22 = .091
7/22 = .318
3/22 = .136
1/22 = .045
0/22 = .000
0/22 = .000
1/22 = .045
1.000
Mode ?
Cumulative R.F.
2/22
4/22
8/22
10/22
17/22
20/22
21/22
21/22
21/22
22/22
13
14
Measure of Dispersion (variability)
Range = largest data value − smallest data value
Is there any difference between
the two samples?
Sample from group I (diet program I):
7, 6, 8, 7, 7
=> mean = (7 + 6 + 8 +7 + 7) / 5 = 35/5 = 7
range of sample I = 8 - 6 = 2
Sample from group II (diet program II):
3, 4, 8, 9, 11
=> mean = (3 + 4 + 8 + 9 + 11) / 5 = 35/5 = 7
range of sample II = 11 - 3 = 8
Does the mother’s diet program
affect the birth weights of babies?
15
16
Example: Birth weights (in lb) of 5 babies born
from a group of women under diet program II.
3, 4, 8, 9, 11 ⇒ mean = x = 7
Data Value Deviation from mean Squared Dev.
xi
Variance and Standard Deviation
Measure the spread of the data around
the center of the data.
17
xi − x
( xi − x ) 2
3
3–7 =–4
16
4
4–7=–3
9
8
8–7=1
1
9
9–7=2
4
11
11 – 7 = 4
16
Total
0
46
Sample Variance = 46/4 = 11.5 lb,
46 / 4 = 3.39 lb.
Sample Standard Deviation =
18
Descriptive Stat - 3
Descriptive Statistics
If n observations are denoted by x1, x2, ..., xn, their variance and
standard deviation are
2
n
s2 =
Sample Variance:
∑ ( xi − x ) 2
i =1
n −1
(unbiased estimator for variance of an infinite population.)
n
s=
Sample Standard Deviation:
Sample Mean:
x=
A Short Cut formula:
∑ ( xi − x ) 2
i =1
n −1
x1 + x2 + ... + xn 1 n
= ∑ xi
n
n i =1
⎛n ⎞
⎜ ∑ xi ⎟
n
2
∑ xi − ⎝ i =1 ⎠
n
s 2 = i =1
n −1
352
291 −
5 = 11.5
=
4
Data, x
x2
3
9
4
16
8
64
9
81
11
121
35
291
19
20
Population Parameters
What is the standard deviation of the
weights of babies from the sample of
mothers who received diet program I?
If N observations are denoted by x1, x2, ..., xn, are all the
observation in a finite population, their mean, µ , variance
σ 2, and standard deviation, σ , are
Data: 7, 6, 8, 7, 7
Population Mean:
µ=
x1 + x2 + ... + xn 1 n
= ∑ xi
N
N i =1
n
s2 = (0+1+1+0+0)/(5-1) = ½
1 = 0.71
2
Does the mother’s diet program
affect the birth weights of babies?
Population Variance:
s=
Diet I: mean = 7,
Diet II: mean = 7,
s = 0.71
s = 3.39
„
„
„
i =1
2
i
N
n
Population Standard Deviation:
σ=
∑ (x − µ)
i =1
21
s measures the spread around the mean.
the larger s is, the more spread out the data
are.
if s = 0, then all the observations must be
equal.
s is strongly influenced by outliers.
N
The Use of Mean and
Standard Deviation
„
„
23
2
i
22
About s (sample standard deviation) :
„
σ2 =
∑ (x − µ)
Describe distribution
Understand the center and the spread
of the distribution
24
Descriptive Stat - 4
Descriptive Statistics
Profit Margin (1972-1981)
American Water Works
x
s
7.61
.68
Birth Weight Data
Unit: oz
Mean,
Which company should
you
7.62
7.39
invest your money?
Brown & Sharpe
Campbell Soup
13.65
1.05
McDonald’s
20.04
1.02
-.98
14.18
Pam American
x
Standard
Deviation, s
Mother Drank
Alcohol
102.2
25
Mother Did not
Drink Alcohol
118.4
20
25
Many distributions can be described by
a mathematical function with specific
parameters, such as mean and
standard deviation.
26
Properties of a symmetric and bell-shaped
(Normal) distribution:
„
„
„
Example: Normal Distribution (Bell-shaped)
„
the distribution is symmetric about it mean (µ),
68% of the area is between µ − σ and µ + σ ,
95% of the area is between µ − 2σ and µ + 2σ ,
99.7% of the area is between µ − 3σ and µ + 3σ .
σ
µ
27
Heart rates for a certain population at a
certain condition follow a bell shape
symmetric distribution with mean 70 and
standard deviation 2.
µ − 3σ
µ
µ + 3σ
28
Chebychev’s inequality
There is at least 1 – (1/k2) of the
data in a data set lie within k
standard deviation of their mean.
What percentage of people in this population
will have heart rates between 66 and 74?
95%
?%
66
70
74
29
30
Descriptive Stat - 5
Descriptive Statistics
Example: Heart rates for asthmatic
patients in a state of respiratory arrest has
a mean of 140 beats per minute and a
standard deviation of 35.5 beats per
minute. What percentage of the population
of this type of patients have heart rates lie
between two standard deviations of the
mean in a state of respiratory arrest?
Heart rates example: mean=144, s.d.=35.5
k=2
75% = 1 − (1/22)
At least 75%
It will be at 75%, because
???
k = 2, and
1 – (1/22) = ¾ = 75%.
69
140 - 2x35.5 = 69
144
211
140 + 2x35.5 = 211
31
What about within three standard deviations?
Heart rates example: mean=144, s.d.=35.5
k=3
2)2)
?% ≈≈11−−(1/3
89%
(1/3
AtAtleast
least89%
?%
32
Coefficient of Variation (C.V.): is the
standard deviation expressed as a
percentage of the mean. It is a unit-free
measure of dispersion. It provides a
measurement for comparing relative
variability of data sets from different scales.
C.V. =
33.5
144 - 3x35.5 = 33.5
144
s
⋅ 100 %
x
246.5
144 + 3x35.5 = 246.5
33
Example: As part of the Berkeley Guidance
Study. The heights (in cm) and weights (in kg)
of 13 girls were measured at age two. The
average height was 86.6 cm with a s.d.= 2.9
and average weight is 12.6 kg with a s.d.= 1.4.
34
Measure of Position
C.V. (height) = (2.9/86.6)100% = 3.3%
C.V. (weight) = (1.4/12.6)100% = 11.1%
Standard Score, Percentile, Quartile
More variability is weight than in height
35
36
Descriptive Stat - 6
Descriptive Statistics
If x is an observation from a
distribution that has mean µ , and
standard deviation σ , the
standardized value of x is,
z-score of x :
z=
If a distribution has a mean 10 and a
s.d. 2, the value 7 has a z-score –1.5.
x −µ
z-score = (7 – 10)/2 = – 1.5.
σ
“µ + 3σ” has a z-score 3, since it is
3 s.d. from mean.
1.5 s.d.
6
8
10
12
14
37
Percentile
(Measure of position)
Find a Data Value Corresponding to a Given Percentile
Step 1: Sort the data.
Step 2: Compute position index c
c = n ⋅ p / 100
38
Example: A sample of number of times visited
class website by 15 students is the following:
27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36,
59, 34, 28. Find the 90th percentile of the data
in this sample.
„
Sol:
n = 15, p = 90.
Ordered data:
20, 25, 25, 27, 28, 31, 33, 34, 36, 37, 44, 50, 59, 85, 86
n = total number of values
p = percentile (If for 90th percentile, p = 90.)
Step 3 (find position):
c = np/100 = 15 x 90 / 100 = 13.5
1) If c is not whole number, round up c to the next
whole number.
2) If c is a whole number, the percentile is at the
position that is halfway between c and c + 1.
Round c to 14. The 14th number in the ordered list
is the 90th percentile and that is 85.
39
Quartiles: (Measure of Position)
40
Example: [odd number of data values] (n = 21)
60,61,63,64,64,65,65,65,66,67,69,71,71,71,72,72,72,72,73,74,75
•
•
The first quartile, Q1, or 25th percentile, is the
median of the lower half of the list of ordered
observations.
The third quartile, Q3, or 75th percentile, is the
median of the upper half of the list of ordered
observations.
Q1 = ?64.5
Median = 69
72
Q3 = ?
Measure of spread:
Interquartile range (IQR) = Q3 − Q1
IQR = 72 – 64.5 = 7.5
41
42
Descriptive Stat - 7
Descriptive Statistics
The five-number summary
Example: [even number of data] (n = 22)
6, 60,61,63,64,64,65,65,65,66,67,69,71,71,71,72,72,72,72,73,74,75
Q1 = 64
?
Median = 68
?
.Minimum value
.Q1
.Median
.Q3
.Maximum value
72
Q3 = ?
Measure of spread:
Interquartile range (IQR) = Q3 − Q1
IQR = 72 - 64 = 8
43
44
With 6 in the data:
Example: (data sheet without outlier “6”)
6, 60,61,63,64,64,65,65,65,66,67,69,71,71,71,72,72,72,72,73,74,75
60,61,63,64,64,65,65,65,66,67,69,71,71,71,72,72,72,72,73,74,75
Q1 = 64
Min = 60, Q1 = 64.5, Median = 69, Q3 = 72, Max = 75.
80
80
70
60
Q3 = 72
40
60
20
50
N=
Median = 68
IQR = 72 - 64 = 8
21
HEIGHT
1
0
45
N=
22
46
HEIGHT
IQR = 72 – 64 = 8; Q1 = 64; Q3 = 72
Inner and outer fences for outliers
•
•
•
The inner fences are located at
a distance of 1.5 IQR below Q1
(lower inner fence = Q1 - 1.5 x IQR )
and at a distance of 1.5 IQR above Q3
(upper inner fence = Q3 + 1.5 x IQR ).
The outer fences are located at
a distance of 3 IQR below Q1
(lower outer fence = Q1 – 3 x IQR )
and at a distance of 3 IQR above Q3
(upper outer fence = Q3 + 3 x IQR ) .
•
47
The inner fences are located at
a distance of 1.5 IQR below Q1
(lower inner fence = 64 - 1.5 x 8 = 52 )
and at a distance of 1.5 IQR above Q3
(upper inner fence = 72 + 1.5 x 8 = 84).
The outer fences are located at
a distance of 3 IQR below Q1
(lower outer fence = 64 – 3 x 8 = 40)
and at a distance of 3 IQR above Q3
(upper outer fence = 72 + 3 x 8 = 96) .
48
Descriptive Stat - 8
Descriptive Statistics
84
UOF:72 + 3 x 8 = 96
96
UIF: 72 + 1.5 x 8 = 84
Outer fence
Inner fence
Inner fence
80
80
IQR
IQR
60
60
Inner fence
Inner fence
52
LIF: 64 - 1.5 x 8 = 52
40
20
40
Q1 = 64; Q3 = 72;
Outer fence
40
LOF: 64 - 3 x 8 = 40
20
IQR = 72 – 64 = 8
1
1
0
N=
0
22
N=
22
HEIGHT
HEIGHT
49
50
Mild and Extreme outliers
Side-by-side Box Plot
80
„
„
Data values falling between the inner and
outer fences are considered mild outliers.
Data values falling outside the outer fences
are considered extreme outliers.
8
14
70
21
18
19
60
HEIGHT
When outliers exist, the whisker extended to
the smallest and largest data values within
the inner fence.
50
N=
sex
51
Boxplot
8
13
Female
Male
52
Remarks:
„
„
3
Find the five-number-summary and
make a boxplot for the following data:
13 72 78 40 50 56 50 52 57
69 130 142 51 52
„
„
„
53
If the distribution of the data is symmetric, then the
mean and median will be about the same.
The five-number summary is best for nonsymmetric
data.
The median and quartiles are not influenced by
outliers.
The mean and standard deviation are most
appropriate to use only if the data are symmetric
because both of these measures are easily
influenced by outliers.
54
Descriptive Stat - 9