1. STA 291 Spring 2008
Lecture 5
Dustin Lueker

2. Measures of Central Tendency
Mean - Arithmetic Average
Median - Midpoint of the observations when they are arranged in increasing order
Notation: Subscripted variables
n = # of units in the sample
N = # of units in the population
x = Variable to be measured
xi = Measurement of the ith unit
Mode - Most frequent value.
STA 291 Spring 2008 Lecture 5

3. Percentiles
The pth percentile (Lp) is a number such that p% of the observations take values below it, and (100-p)% take values above it
50th percentile = median
25th percentile = lower quartile
75th percentile = upper quartile
The index of Lp
(n+1)p/100
STA 291 Spring 2008 Lecture 5

4. Quartiles 25th percentile 75th percentile lower quartile Q1
(approximately) median of the observations below the median
75th percentile
upper quartile Q3
(approximately) median of the observations above the median
STA 291 Spring 2008 Lecture 5

5. Example Find the 25th percentile of this data set
{3, 7, 12, 13, 15, 19, 24}
STA 291 Spring 2008 Lecture 5

6. Interpolation Use when the index is not a whole number
Want to go closest index lower then go the distance of the decimal towards the next number
If the index is found to be 5.4 you want to go to the 5th value then add .4 of the value between the 5th value and 6th value
In essence we are going to the 5.4th value
STA 291 Spring 2008 Lecture 5

7. Example Find the 40th percentile of the same data set
{3, 7, 12, 13, 15, 19, 24}
Must use interpolation
STA 291 Spring 2008 Lecture 5

8. Data Summary Five Number Summary Example Minimum Lower Quartile Median
Upper Quartile
Maximum
Example
minimum=4
Q1=256
median=530
Q3=1105
maximum=320,000.
What does this suggest about the shape of the distribution?
STA 291 Spring 2008 Lecture 5

9. Interquartile Range (IQR)
The Interquartile Range (IQR) is the difference between upper and lower quartile
IQR = Q3 – Q1
IQR = Range of values that contains the middle 50% of the data
IQR increases as variability increases
Murder Rate Data
Q1= 3.9
Q3 = 10.3
IQR =
STA 291 Spring 2008 Lecture 5

10. Box Plot Displays the five number summary (and more) graphical
Consists of a box that contains the central 50% of the distribution (from lower quartile to upper quartile)
A line within the box that marks the median,
And whiskers that extend to the maximum and minimum values
This is assuming there are no outliers in the data set
STA 291 Spring 2008 Lecture 5

11. Outliers An observation is an outlier if it falls
more than 1.5 IQR above the upper quartile or
more than 1.5 IQR below the lower quartile
STA 291 Spring 2008 Lecture 5

12. Box Plot
Whiskers only extend to the most extreme observations within 1.5 IQR beyond the quartiles
If an observation is an outlier, it is marked by an x, +, or some other identifier
STA 291 Spring 2008 Lecture 5

13. Example Values Create a box plot Min = 148 Q1 = 158 Median = Q2 = 162
Max = 204
Create a box plot
STA 291 Spring 2008 Lecture 5

14. Mode Value that occurs most frequently Special Cases
Does not need to be near the center of the distribution
Not really a measure of central tendency
Can be used for all types of data (nominal, ordinal, interval)
Special Cases
Data Set
{2, 2, 4, 5, 5, 6, 10, 11}
Mode =
{2, 6, 7, 10, 13}
STA 291 Spring 2008 Lecture 5

15. Mean vs. Median vs. Mode Mean Median Mode
Interval data with an approximately symmetric distribution
Median
Interval or ordinal data
Mode
All types of data
STA 291 Spring 2008 Lecture 5

16. Mean vs. Median vs. Mode Mean is sensitive to outliers
Median and mode are not
Why?
In general, the median is more appropriate for skewed data than the mean
In some situations, the median may be too insensitive to changes in the data
The mode may not be unique
STA 291 Spring 2008 Lecture 5

17. Example “How often do you read the newspaper?” Identify the mode
Response
Frequency
every day
969
a few times a week
452
once a week
261
less than once a week
196
Never 76
TOTAL
1954
Identify the mode
Identify the median response
STA 291 Spring 2008 Lecture 5

18. Measures of Variation Statistics that describe variability
Two distributions may have the same mean and/or median but different variability
Mean and Median only describe a typical value, but not the spread of the data
Range
Variance
Standard Deviation
Interquartile Range
All of these can be computed for the sample or population
STA 291 Spring 2008 Lecture 5

19. Range Difference between the largest and smallest observation
Very much affected by outliers
A misrecorded observation may lead to an outlier, and affect the range
The range does not always reveal different variation about the mean
STA 291 Spring 2008 Lecture 5

20. Example Sample 1 Sample 2 Smallest Observation: 112
Largest Observation: 797
Range =
Sample 2
Smallest Observation:
Largest Observation:
STA 291 Spring 2008 Lecture 5

21. Variance and Standard Deviation
Sample
Variance
Standard Deviation
Population
STA 291 Spring 2008 Lecture 5

22. Deviations
The deviation of the ith observation xi from the sample mean is the difference between them,
Sum of all deviations is zero
Therefore, we use either the sum of the absolute deviations or the sum of the squared deviations as a measure of variation
STA 291 Spring 2008 Lecture 5

23. Sample Variance
Variance of n observations is the sum of the squared deviations, divided by n-1
STA 291 Spring 2008 Lecture 5

24. Example Observation Mean Deviation Squared 1 3 4 7 10
Sum of the Squared Deviations
n-1
Sum of the Squared Deviations / (n-1)
STA 291 Spring 2008 Lecture 5

25. Interpreting Variance
About the average of the squared deviations
“average squared distance from the mean”
Unit
Square of the unit for the original data
Difficult to interpret
Solution
Take the square root of the variance, and the unit is the same as for the original data
Standard Deviation
STA 291 Spring 2008 Lecture 5

26. Properties of Standard Deviation
s = 0 only when all observations are the same
If data is collected for the whole population instead of a sample, then n-1 is replaced by n
s is sensitive to outliers
STA 291 Spring 2008 Lecture 5

27. Empirical Rule
If the data is approximately symmetric and bell-shaped then
About 68% of the observations are within one standard deviation from the mean
About 95% of the observations are within two standard deviations from the mean
About 99.7% of the observations are within three standard deviations from the mean
STA 291 Spring 2008 Lecture 5

28. Example
SAT scores are scaled so that they have an approximate bell-shaped distribution with a mean of 500 and standard deviation of 100
About 68% of the scores are between
About 95% of the scores are between
If you have a score above 700, you are in the
top %
What percentile would this be?
STA 291 Spring 2008 Lecture 5

29. Example
According to the National Association of Home Builders, the U.S. nationwide median selling price of homes sold in 1995 was $118,000
Would you expect the mean to be larger, smaller, or equal to $118,000?
Which of the following is the most plausible value for the standard deviation?
(a) –15,000, (b) 1,000, (c) 45,000, (d) 1,000,000
STA 291 Spring 2008 Lecture 5

30. Population Parameters and Sample Statistics
Population mean and population standard deviation are denoted by the Greek letters μ (mu) and σ (sigma)
They are unknown constants that we would like to estimate
Sample mean and sample standard deviation are denoted by and s
They are random variables, because their values vary according to the random sample that has been selected
STA 291 Spring 2008 Lecture 5