Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures
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Introduction to Biostatistics(Pubhlth 540)
Lecture 3: Numerical Summary
Measures
Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material
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Reading/Home work
• -See WEB site
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For after all, what is man in nature? A Nothing in relation to the infinite, All in relation to nothing, A central point between nothing
and all,And infinitely far from understanding
either.
Blaise Pascal, (1623-1662) Pensees (1660)
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2.30x =
Let x represent FEV1 in liters
Example: FEV per second in 13 adolescents with asthma
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2.30
2.15
x
x
=
=
Let x represent FEV1 in liters
Example: FEV per second in 13 adolescents with asthma
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1
2
2.30
2.15
x
x
=
=
Let x represent FEV1 in liters
Example: FEV per second in 13 adolescents with asthma
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1
2
3
2.30
2.15
3.50
x
x
x
=
=
=
Let x represent FEV1 in liters
Example: FEV per second in 13 adolescents with asthma
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1
2
3
4
5
6
2.30
2.15
3.50
2.60
2.75
2.82
x
x
x
x
x
x
=
=
=
=
=
=
Let x represent FEV1 in liters
Example: FEV per second in 13 adolescents with asthma
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1
2
3
4
5
6
2.30
2.15
3.50
2.60
2.75
2.82
x
x
x
x
x
x
=
=
=
=
=
=
Let x represent FEV1 in liters
7
8
9
10
11
12
13
4.05
2.25
2.68
3.00
4.02
2.85
3.38
x
x
x
x
x
x
x
=
=
=
=
=
=
=
Example: FEV per second in 13 adolescents with asthma
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Measures of central tendency
•Mean
•Median
•Mode
•Population Parameters•Sample Statistics
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Measures of central tendency
•Population Parameters
N
1
1Population Mean
N ss
x
m=
=
=
å
( )N
2
1
2
1Population Variance
N ss
x m
s=
= -
=
å
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n
1
1Sample Mean
n ii
x=
= å
Measures of central tendency: Mean
2.3, 2.15, 3.50, 2.60, 2.75, 2.82, 4.05,2.25, 2.68, 3.00, 4.02, 2.85 (n=13)
n
1
Sum 38.35ii
x=
= =å.
Mean = x .= =38 35
2 9513
1 2Sample Numbers: , ,..., nx x x
Example: FEV per second in 13 adolescents with asthma
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If we collect a man's urine during twenty four hours and mix all this urine to analyze the average, we get an analysis of a urine which simply does not exist; for urine when fasting, is different from urine during digestion. A startling instance of this kind was invented by a physiologist who took urine from a railroad station urinal where people of all nations passed, and who believed he could thus present an analysis of average European urine!
Claude Bernard (1813-1878)
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Variable Mean
Mother’s age 26.4 years
Approx 4 million singleton births, 1991 :
Mean: Examples
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Variable Mean
Mother’s age 26.4 years
Gestational age 39.15 weeks
Approx 4 million singleton births, 1991 :
Mean: Examples
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Variable Mean
Mother’s age 26.4 years
Gestational age 39.15 weeks
Birth weight 3358.6 grams
Approx 4 million singleton births, 1991 :
Mean: Examples
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Variable Mean
Mother’s age 26.4 years
Gestational age 39.15 weeks
Birth weight 3358.6 grams
Weight gain* 30.4 lbs
Approx 4 million singleton births, 1991 :
Mean: Examples
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Variable Mean
Mother’s age 26.4 years
Gestational age 39.15 weeks
Birth weight 3358.6 grams
Weight gain* 30.4 lbs
Approx 4 million singleton births, 1991 :
Of 31,417 singleton births resulting in death :
Survival 49.4 days
Mean: Examples
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years
26.4 years
Mean: Properties
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Note what happens when one number,4.02 say, becomes large, say 40.2 :
2.3, 2.15, 3.50, 2.60, 2.75, 2.82, 4.05, 2.25, 2.68, 3.00, 40.2, 2.85
Mean = x =5.73
(versus 2.95, from before)
Mean is sensitive to every observation,it is not robust.
Mean: Properties
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Measures of central tendency: Median
More robust, but not sensitive enough.
Definition: At least 50% of the observations are greater than or equal to the median, and at least 50% of the observations are less than or equal to the median.
2.15, 2.25, 2.30 --- median = 2.25
2.15, 2.25, 2.30, 2.60 ---12(2.25 + 2.30) = 2.275median =
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Variable Mean Median
Mom’s age (yrs)
26.4 25
Gest. Age (wks) 39.2 39
Birth weight (gms)
3359 3374
Weight gain (lbs)
30.4 30
Survival (days) 49.4 7
Singleton births, 1991 :
Comparing mean and median
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Mean = 3359 Median = 3374
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Mean = 30.4 Median = 30
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Mortality in the fi rst year of baby's lif e
(f or those who die in their fi rst year)
0.00
0.10
0.20
0.30
0.40
0 60 121 182 244 305
(survival days)
Prop
orti
on
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Mortality in the first year of baby's life(for those who die in their first year)
0.00
0.00
0.01
0.10
1.00
0 31 60 91 121 152182 213 244274 305335
(survival days)
Prop
ortio
n
Mean = 49.4 Median=7
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When to use mean or median:
Use both by all means.
Mean performs best when we have asymmetric distribution with thin tails.
If skewed, use the median.
Remember: the mean follows the tail.
Comparing mean and median
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Mode
• Mode is defined as the observation that occurs most frequently
• When the distribution is symmetric, all three measures of central tendency are equal
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Comparing mean, median and mode
Bimodal distribution
Modes
Mean, Median
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•Range:•Simple to calculate•Very sensitive to extreme observations
•Inter Quartile Range (IQR) •More robust than the range
•Variance (Standard Deviation):
•Quantifies the amount of variability around the mean
Measures of spread
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Variable Min Max Range
Mom’s age 10 49 39
Gest. Age 17 47 30
Birth weight
227 8164 7937
Weight gain
0 98 98
Survival 0 363 363
Singleton births, 1991 :
Measures of spread: Range
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FEV1
2.30-0.65
2.15 -0.80
3.50 0.55
2.60 -0.35
2.75 -0.20
2.82 -0.13
4.05 1.10
2.25 -0.70
2.68 -0.27
3.00 0.05
4.02 1.07
2.85 -0.10
3.38 0.43
( )jx x-
Measures of spread: Variance
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FEV1
2.30-0.65
2.15 -0.80
3.50 0.55
2.60 -0.35
2.75 -0.20
2.82 -0.13
4.05 1.10
2.25 -0.70
2.68 -0.27
3.00 0.05
4.02 1.07
2.85 -0.10
3.38 0.43
Total 0.00
( )jx x-
Measures of spread: Variance
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FEV1
2.30-0.65
0.423
2.15 -0.80 0.640
3.50 0.55 0.303
2.60 -0.35 0.123
2.75 -0.20 0.040
2.82 -0.13 0.169
4.05 1.10 1.210
2.25 -0.70 0.490
2.68 -0.27 0.073
3.00 0.05 0.003
4.02 1.07 1.145
2.85 -0.10 0.010
3.38 0.43 0.185
Total 0.00 4.66
( )jx x- 2( )jx x-
Measures of spread: Variance
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2
1
1Sample Variance = ( )
n-10
n
ii
x x=
-
³
å
e.g.
24.660.39liters
12= =
Measures of spread: Variance
2
1
1Population Variance = ( )
N
N
ss
x m=
-å
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Standard deviation = + Variance
e.g.
0.39
0.62liters
=
=
Measures of spread: Variance
Standard deviation takes on the same unit as the mean
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Empirical Rule:
If dealing with a unimodal andsymmetric distribution, then
Mean ± 1 sd covers approx 67% obs.
Mean ± 3 sd covers approx all obs
Mean ± 2 sd covers approx 95% obs
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24 67%
Table of x ± k s.d.s
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24 67%2 14.72 38.08
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24 67%2 14.72 38.08 95%
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24 67%2 14.72 38.08 95%3 8.88 43.92
Table of x ± k s.d.s
Variance & Standard deviation
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
1 20.56 32.24 67%2 14.72 38.08 95%3 8.88 43.92 all
Table of x ± k s.d.s
Characterizing a symmetric, unimodal distribution – mean,
SD
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years20.56 32.4
Area = 0.6475
Characterizing a symmetric, unimodal distribution – mean,
SD
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years14.72 38.08
Area = 0.963
Characterizing a symmetric, unimodal distribution – mean,
SD
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Mother’s age: mean = 26.4 yrs s.d. = 5.84 yrs
kleft limit
right limit
Emp.
Actual
1 20.56 32.24 67% 64.75%
2 14.72 38.08 95% 96.3%
3 8.88 43.92 all 99.89%
Table of x ± k s.d.s
Characterizing a symmetric, unimodal distribution – mean,
SD
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Chebychev’s Inequality
Table of x ± k s.d.s
Proportion is at least 1-1/k2
(true for any distribution.)
Characterizing a distribution – Chebychev’s inequality
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Chebychev’s Inequality
k 1/k2
1 1
2 0.25
3 0.11
Table of x ± k s.d.s
Proportion is at least 1-1/k2
(true for any distribution.)
Characterizing a distribution – Chebychev’s inequality
![Page 52: 1 Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures Acknowledgement: Thanks to Professor Pagano (Harvard School of Public.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d445503460f94a20cff/html5/thumbnails/52.jpg)
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Chebychev’s Inequality
k 1/k2 1-1/k2
1 1 0
2 0.25 0.75
3 0.11 0.89
Table of x ± k s.d.s
Proportion is at least 1-1/k2
(true for any distribution.)
Characterizing a distribution – Chebychev’s inequality
![Page 53: 1 Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures Acknowledgement: Thanks to Professor Pagano (Harvard School of Public.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d445503460f94a20cff/html5/thumbnails/53.jpg)
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Chebychev’s Inequality
k 1/k2 1-1/k2 Emp.
1 1 0 67%2 0.25 0.75 95%3 0.11 0.89 all
Table of x ± k s.d.s
Proportion is at least 1-1/k2
(true for any distribution.)
Characterizing a distribution – Chebychev’s inequality
![Page 54: 1 Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures Acknowledgement: Thanks to Professor Pagano (Harvard School of Public.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d445503460f94a20cff/html5/thumbnails/54.jpg)
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Chebychev’s Inequality
k 1/k2 1-1/k2 Emp.
Actual
1 1 0 67%64.75
%2 0.25 0.75 95% 96.3%
3 0.11 0.89 all99.89
%
Table of x ± k s.d.s
Proportion is at least 1-1/k2
(true for any distribution)
Characterizing a distribution – Chebychev’s inequality
![Page 55: 1 Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures Acknowledgement: Thanks to Professor Pagano (Harvard School of Public.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d445503460f94a20cff/html5/thumbnails/55.jpg)
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Summary• Distributions can be described using:
– Measures of central tendency– Measures of dispersion
• Measures of central tendency: – Mean, Median, Mode
• Measures of dispersion: – Range, IQR, Variance, Standard Deviation
• Characterizing distributions: – Chebyshev’s inequality– Empirical rule for symmetric, unimodal
distributions
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Questions
• In a certain real estate market, the average price of a single family home was $325,000 and the median price was $225,000. Percentiles were computed for this distribution. Is the difference between the 90th and 50th percentile likely to be bigger than, about the same as, or less than the difference between the 50th and 10th percentile? Explain briefly.
http://www.stat.berkeley.edu/users/rice/Stat2/Chapt4.pdf
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Questions
http://www.stat.berkeley.edu/users/rice/Stat2/Chapt4.pdf
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Questions
• 1. The average high temperature for Minneapolis is closest to (a) 45 degrees (b) 60 degrees (c) 75 degrees (d) 85 degrees
• 2. The SD of the high temperatures for Minneapolis is closest to (a) 1 degree (b) 3 degrees (c) 5 degrees (d) 20 degrees
• 3. The average high temperature for Minneapolis is --------- _the average high temperature for Belle Glade. (a) at least ten degrees less than (b) about the same as (c) at least ten degrees higher than
• 4. The average high temperature for Minneapolis is --------_the average high temperature for Olga. (a) at least ten degrees less than (b) about the same as (c) at least ten degrees higher than
• 5. The SD of the high temperatures for Minneapolis is -------- the SD of the high temperatures for Belle Glade. (a) about half of (b) about the same as (c) about twice
http://www.stat.berkeley.edu/users/rice/Stat2/Chapt4.pdf