1. A primer in Biostatistics
Christina M. Ramirez
UCLA Department of Biostatistics

2. Statistics Data Collection Summarizing Data Interpreting Data
Drawing Conclusions from Data

3. Population
The set of data (numerical or otherwise) corresponding to the entire collection of units about which information is sought
Example: Unemployment - Status of ALL employable people (employed, unemployed) in the country.

4. Sample
A subset of the population data that are actually collected in the course of a study.
Example: Unemployment - Status of the 1000 employable people interviewed.

5. Population vs. Sample Population Sample
In most studies, it is difficult to obtain information from the entire population. We rely on samples to make estimates or inferences related to the population.

6. Descriptive statistics
Describing data with numbers:
measures of location

7. What to describe?
What is the “location” or “center” of the data? (“measures of location”)
Mean
Median
Mode
How do the data vary? (“measures of variability”)
Range
Interquartile Range
Variant

8. Mean Another name for average.
Appropriate for describing measurement data.
Seriously affected by unusual values called “outliers”.

9. Calculating Sample Mean
Add up all of the data points and divide by the number of data points.
Example:
Number of drinks/day:
Sample Mean = ( )/5 = 3.6

10. Median Another name for 50th percentile.
Appropriate for describing measurement data.
“Robust to outliers,” that is, not affected much by unusual values.

11. Calculating Sample Median
Order data from smallest to largest.
If odd number of data points, the median is the middle value.
Number of drinks/day:
Ordered Data:
Median

12. Mode The value that occurs most frequently.
One data set can have many modes.
Appropriate for all types of data, but most useful for categorical data or discrete data with only a few number of possible values.
Example: Number of eyes affected with cataracts in 70 year olds: 0, 1, 2.

13. The most appropriate measure of location depends on …
the shape of the data’s distribution.

14. Most appropriate measure of location
Depends on whether or not data are “symmetric” or “skewed”.
Depends on whether or not data have one (“unimodal”) or more (“multimodal”) modes.

15. Symmetric and Unimodal

16. Symmetric and Bimodal

17. Symmetric and Bimodal

18. Skewed Right

19. Choosing Appropriate Measure of Location
If data are symmetric, the mean, median, and mode will be approximately the same.
If data are multimodal, report the mean, median and/or mode for each subgroup.
If data are skewed, report the median.

20. Descriptive statistics
Describing data with numbers: measures of variability
Range
Interquartile range
Variance and standard deviation

21. Range The difference between largest and smallest data point.
Highly affected by outliers.
Best for symmetric data with no outliers.

22. What is the range?

23. Interquartile range
The difference between the “third quartile” (75th percentile) and the “first quartile” (25th percentile). So, the “middle-half” of the values.
IQR = Q3-Q1
Robust to outliers or extreme observations.
Works well for skewed data.

24. Interquartile range IQR = 3.4675 - 2.6725 = 0.795
Descriptive Statistics
Variable N Mean Median TrMean StDev SE Mean
GPA
Variable Minimum Maximum Q Q3
GPA
IQR = = 0.795

25. Variance 1. Find difference between each data point and mean.
2. Square the differences, and add them up.
3. Divide by one less than the number of data points.

26. Variance
If measuring variance of population, denoted by 2 (“sigma-squared”).
If measuring variance of sample, denoted by s2 (“s-squared”).
Measures average squared deviation of data points from their mean.
Highly affected by outliers. Best for symmetric data.

27. Standard deviation
Sample standard deviation is square root of sample variance, and so is denoted by s.
Units are the original units.
Measures average deviation of data points from their mean.
Also, highly affected by outliers.

28. What is the variance or standard deviation?

29. Variance or standard deviation
Sex N Mean Median TrMean StDev SE Mean
female
male
Sex Minimum Maximum Q Q3
female
male
Females: s = kph and s2 = = kph2
Males: s = kph and s2 = = kph2

30. The most appropriate measure of variability depends on …
the shape of the data’s distribution.

31. Choosing Appropriate Measure of Variability
If data are symmetric, with no serious outliers, use range and standard deviation.
If data are skewed, and/or have serious outliers, use IQR.

32. Probability
The “p” in p-value

33. Examples: Coin Flips Flips #(Flips) #(Heads) P(H)
Ben 4, ,
Christina 24, ,
Roger 10, ,

34. Randomness, Independence, Multiplication Rule
Probability Concepts
Randomness, Independence, Multiplication Rule

35. Thought Question 1
What does it mean to say that a deck of cards is “randomly” shuffled?
Every ordering of the cards is equally likely
There are 8 followed by 67 zeros possible orderings of a 52 card deck
Every card has the same probability to end up in any specified location

36. The question continued
A 52 card deck is randomly shuffled
How often will the tenth card down from the top be a Club?
1/4 of the time
Every card has the same chance to end up 10th. There are 13 clubs and 13 / 52 = 1/4

37. More of the question Deck had three cards - labeled A, B, C
After a random shuffle, cards are turned over one at a time.
How often is the A card the second card that’s turned over?
1/3 : each card had the same chance to end up in a specific position

38. Thought Question 2
A fair die is rolled many times. How often will a “1” be the result?
About 1/6 of the time, but there will be some sampling error
How does increasing the number of rolls affect the difference between sample fraction of “1”’s and 1/6?
Difference likely to get smaller as n increases since margin of error goes down

39. Does a prior event matter?
A fair coin is flipped four times.
First three flips are heads
What’s the probability that the fourth flip is heads?
1/2 assuming flips are independent
Results of first three flips don’t matter

40. Does prior event matter?
Ten cards are drawn without replacement from 52 card deck.
2 Aces are among these 10 cards
What’s the probability the eleventh card is an Ace?
2/42 = 1/21
After ten draws, 42 cards remain, 2 of them are Aces