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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. Reading/Home work. -See WEB site. For after all, what is man in nature?

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introduction to biostatistics pubhlth 540 lecture 3 numerical summary measures
Introduction to Biostatistics(Pubhlth 540) Lecture 3: Numerical Summary Measures

Acknowledgement: Thanks to Professor Pagano

(Harvard School of Public Health) for lecture material

reading home work
Reading/Home work
  • -See WEB site
slide3
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)

slide10

Measures of central tendency

  • Population Parameters
  • Sample Statistics
  • Mean
  • Median
  • Mode
slide11

Measures of central tendency

  • Population Parameters
slide12

Measures of central tendency: Mean

Example: FEV per second in 13 adolescents with asthma

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)

slide13

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)

slide14

Mean: Examples

Approx 4 million singleton births, 1991 :

slide15

Mean: Examples

Approx 4 million singleton births, 1991 :

slide16

Mean: Examples

Approx 4 million singleton births, 1991 :

slide17

Mean: Examples

Approx 4 million singleton births, 1991 :

slide18

Mean: Examples

Approx 4 million singleton births, 1991 :

Of 31,417 singleton births resulting

in death :

slide19

Mean: Properties

26.4 years

years

slide20

Mean: Properties

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

(versus 2.95, from before)

Mean is sensitive to every observation,

it is not robust.

slide21

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 ---

(2.25 + 2.30) = 2.275

median =

slide22

Comparing mean and median

Singleton births, 1991 :

slide28

Comparing mean and median

When to use mean or median:

Use both by all means.

Mean performs best when we have a

symmetric distribution with thin tails.

If skewed, use the median.

Remember: the mean follows the tail.

slide29
Mode
  • Mode is defined as the observation that occurs most frequently
  • When the distribution is symmetric, all three measures of central tendency are equal
comparing mean median and mode
Comparing mean, median and mode

Bimodal distribution

Mean, Median

Modes

slide31

Measures of spread

  • 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
slide32

Measures of spread: Range

Singleton births, 1991 :

slide37

Measures of spread: Variance

Standard deviation takes on the same unit as the mean

slide38

Variance & Standard deviation

Empirical Rule:

If dealing with a unimodal and

symmetric distribution, then

Mean ± 1 sd covers approx 67% obs.

Mean ± 2 sd covers approx 95% obs

Mean ± 3 sd covers approx all obs

slide39

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide40

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide41

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide42

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide43

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide44

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide45

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide46

Characterizing a symmetric, unimodal distribution – mean, SD

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

slide49

Characterizing a symmetric, unimodal distribution – mean, SD

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± ks.d.s

slide50

Characterizing a distribution – Chebychev’s inequality

Chebychev’s Inequality

Table of

± k s.d.s

Proportion is at least 1-1/k2

(true for any distribution.)

slide51

Characterizing a distribution – Chebychev’s inequality

Chebychev’s Inequality

Table of

± k s.d.s

Proportion is at least 1-1/k2

(true for any distribution.)

slide52

Characterizing a distribution – Chebychev’s inequality

Chebychev’s Inequality

Table of

± k s.d.s

Proportion is at least 1-1/k2

(true for any distribution.)

slide53

Characterizing a distribution – Chebychev’s inequality

Chebychev’s Inequality

Table of

± k s.d.s

Proportion is at least 1-1/k2

(true for any distribution.)

slide54

Characterizing a distribution – Chebychev’s inequality

Chebychev’s Inequality

Table of

± k s.d.s

Proportion is at least 1-1/k2

(true for any distribution)

summary
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
questions
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

questions1
Questions

http://www.stat.berkeley.edu/users/rice/Stat2/Chapt4.pdf

questions2
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

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