Christina M. Ramirez
UCLA Department of Biostatistics

Statistics

Data Collection

Summarizing Data

Interpreting Data

Drawing Conclusions from Data

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.

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.

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.

Describing data with numbers: measures of location

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

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

Calculating Sample Mean
Add up all of the data points and divide by the number of data points.
Example:
Number of drinks/day: 2 8 3 4 1
Sample Mean = (2+8+3+4+1)/5 = 3.6

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

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: 2 8 3 4 1
Ordered Data: 1 2 3 4 8
Median

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.

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

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.

Symmetric and Unimodal
20
10
0
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
GPAs

Symmetric and Bimodal

Symmetric and Bimodal

Skewed Right
Number of Music CDs of Spring 1998 Stat 250 Students
20
10
0
0 100 200 300
Number of Music CDs
400

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.

Describing data with numbers: measures of variability
•
•
•
Range
Interquartile range
Variance and standard deviation

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

What is the range?
20
GPAs of Spring 1998 Stat 250 Students
10
0
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
GPA

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.

Interquartile range
Descriptive Statistics
Variable N Mean Median TrMean StDev SE Mean
GPA 92 3.0698 3.1200 3.0766 0.4851 0.0506
Variable Minimum Maximum Q1
GPA 2.0200 3.9800 2.6725
Q3
3.4675
IQR = 3.4675 - 2.6725 = 0.795

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.

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

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.

Sex male
What is the variance or standard deviation?
Fastest Ever Driving Speed female
120 170
KPH
220 270

Variance or standard deviation
Sex N Mean Median TrMean StDev SE Mean female 126 152.05 150.00 151.39 18.86 1.68 male 100 177.98 183.33 176.04 28.98 2.90
Sex Minimum Maximum Q1 Q3 female 108.33 200.00 141.67 163.75
male 125.00 270.00 158.33 197.92
Females: s = 18.86 kph and s 2 = 18.86
2 = 355.7 kph 2
Males: s = 28.98 kph and s 2 = 28.98
2 = 839.8 kph 2

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

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.

The “p” in p-value

Examples: Coin Flips
Flips
Ben
Christina
Roger
#(Flips)
4,040
24,000
10,000
#(Heads)
2,048
12,012
5,067
P(H)
0.5069
0.5005
0.5067


Randomness, Independence,
Multiplication Rule

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

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

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

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

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

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