1. Numerical Descriptive Measures

2. Motivation What is the “average consumer” exactly?
Why is it that if an average yield on an investment (e.g. mutual fund) is 28% that I’ve lost money?
At Morning Brew people arrive on average 1 per min and it takes typically 1 min to serve them, why is it that if I staff the register with 1 person people complain that the lines are too long and often leave before purchasing something?
If I plan our payables based on an average daily receivables of $7,000 why have I gone bankrupt?
Why do students always want to know what the average on the exam was?

3. Summary Measures Describing Data Numerically Central Tendency
Quartiles
Variation
Shape
Arithmetic Mean
Range
Skewness
Median
Interquartile Range
Kurtosis
Mode
Standard Deviation
Coefficient of Variation

4. Measures of Central Tendency
Overview
Central Tendency
Arithmetic Mean
Median
Mode
“Balance Point” of data. Usually not in data set.
Midpoint of ranked values. In an ordered array, the median is the “middle” number (50% above, 50% below). May by in data set.
Most frequently observed value (multiple or may not exist esp. continuous data). Always in data set.

5. Which One to Use?
Mean is generally used, unless extreme values (outliers) exist
Median is often used, since the median is not sensitive to extreme values.
Mode is rarely used because there may be no mode, and there may be several modes
…. 500
Mean = 58
Median = 3
Mode = 2, 4
Mean = 3
Median = 3
Mode = 2, 4

6. Quartiles
Quartiles split the ranked data into 4 segments with an equal number of values per segment
25%
25%
25%
25%
Q Q Q3
25th Percentile th percentile 75th percentile
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile

7. Class Exercise: Tendency & Histograms
mode 5
median
4.9 Q1
4.4 Q2
4.9 Q3
5.1 Q4 6
mean
Place the mean, median, mode on the histogram. What do you see?

8. Example
Median home prices usually are reported for a region – less sensitive to outliers
Example: Five houses on a hill by the beach
Mean: ($3,000,000/5)
= $600,000
Median: middle value of ranked data = $300,000
Mode: most frequent value = $100,000
House Prices: $2,000,000
500, , , ,000
Sum $3,000,000
Think about this: Which average best helps you decide what to offer for a house? How about set selling price? What other considerations might there be?

9. What’s The Difference?
Mean $600,000

10. Coefficient of Variation
Measures of Variation
Measures of variation give information on the spread or variability of the data values.
More variable
Less variable
Range
Interquartile
Range
Variation
Standard Deviation
Same center,
different variation
Coefficient of Variation

11. Range and Interquartile Range
Example:
Median
(Q2) X Q1 Q3
minimum X
maximum
25% % % %
Range = 70 – 12 = 58
Range
= Xmaximum– Xminimum
Simplest measure
Sensitive to outliers
Interquartile range
= 57 – 30 = 27
Interquartile Range
= Q3 – Q1
Measure Middle 50%
Eliminate outliers Problem

12. Disadvantages of the Range
Range ignores the way in which data are distributed
Range = = 7
IQR = 11 – 8 = 3
Range = = 7
IQR = 11 – 10 = 1
Range is also sensitive to outliers
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5
Range = = 4 IQR = 2 – 1 = 1
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,120
Range = = IQR = 2 – 1 = 1

13. Standard Deviation Most commonly used measure of variation
Each value in the data set is used in the calculation
Shows variation from the mean
Values far from the mean are given extra weight (because deviations from the mean are squared)
Has the same units as the original data
Sample standard deviation:

14. Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Can be used to compare two or more sets of data measured in different units

15. Comparing Variation Standard Deviations
Data A
Mean = 15.5
SD = 3.338
Data B
Mean = 15.5
SD = 0.926
Data C
Mean = 15.5
SD = 4.567

16. Comparing Variation Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Both stocks have the same standard deviation, but stock B is less variable relative to its price

17. Shape of a Distribution
Describes how data are distributed
Measures of shape
Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean

18. Normal or Bell-shaped Curve
Normal Exact
Normal Ok
Skewness = 0
Kortosis = 0
-1< SK <+1
-1< K <+1
Normal or Bell-shaped Curve
Mean and Standard Deviation define what a normal curve looks like
Example:
Mean: μ =0
Standard Deviation: σ=15
IQR=20
Right Skewed SK >0
Left Skewed SK <0
Peaked K >0
Flat K <0
Box-and-Whisker
Approx. 50%
Approx. 68%
Approx. 95%
Almost 100%
K>0
K<0
SK>0
SK<0
Mean, Mode, Median
Peaked Flat
Right skewed Left skewed

19. The Empirical Rule
If the data distribution is approximately bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
68%

20. The Empirical Rule
contains about 95% of the values in the population or the sample
contains about 99.7% of the values in the population or the sample
95%
99.7%

21. Exploratory Data Analysis
5-number summary:
Box-and-Whisker Plot: A Graphical display of 5-number summary. It shows both Central Tendency, Variation, and Shape of the numerical variable.
Minimum -- Q1 -- Median -- Q3 -- Maximum
25% % % %
Central Tendency
Variation

22. Shape and B-n-W Plot Left-Skewed Symmetric Right-Skewed Q1 Q2 Q3 Q1 Q2

23. Shape and B-n-W Plot Cont’d
Left-Skewed
Symmetric
Right-Skewed
Peaked
Flat