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Introduction to Biostatistics (Pubhlth 540) Lecture 3: Numerical Summary Measures

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Acknowledgement: Thanks to Professor Pagano

(Harvard School of Public Health) for lecture material

- -See WEB site

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)

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Example: FEV per second in 13 adolescents with asthma

Let x represent FEV1 in liters

Measures of central tendency

- Population Parameters
- Sample Statistics

- Mean
- Median
- Mode

Measures of central tendency

- Population Parameters

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)

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)

Mean: Examples

Approx 4 million singleton births, 1991 :

Mean: Examples

Approx 4 million singleton births, 1991 :

Mean: Examples

Approx 4 million singleton births, 1991 :

Mean: Examples

Approx 4 million singleton births, 1991 :

Mean: Examples

Approx 4 million singleton births, 1991 :

Of 31,417 singleton births resulting

in death :

Mean: Properties

26.4 years

years

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.

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 =

Comparing mean and median

Singleton births, 1991 :

Mean = 3359 Median = 3374

Mean = 30.4 Median = 30

Mean = 49.4 Median=7

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.

- Mode is defined as the observation that occurs most frequently
- When the distribution is symmetric, all three measures of central tendency are equal

Bimodal distribution

Mean, Median

Modes

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

Measures of spread: Range

Singleton births, 1991 :

Measures of spread: Variance

Measures of spread: Variance

Measures of spread: Variance

Measures of spread: Variance

e.g.

Measures of spread: Variance

Standard deviation takes on the same unit as the mean

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

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

Variance & Standard deviation

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± k s.d.s

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

Characterizing a symmetric, unimodal distribution – mean, SD

Area = 0.6475

20.56

32.4

years

Characterizing a symmetric, unimodal distribution – mean, SD

Area = 0.963

14.72

38.08

years

Characterizing a symmetric, unimodal distribution – mean, SD

Mother’s age: mean = 26.4 yrs

s.d. = 5.84 yrs

Table of

± ks.d.s

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.)

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.)

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.)

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.)

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)

- 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

- 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

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

- 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