1. Data Characterization
Chapter 3
Data Characterization
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2. Types of Data Measurements
Measurements of Center and Location
Measurements of Variation ?
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3. Measurements for Population and Sample
In general, we use the same set of measurements for both population and sample
Population Parameters: numerical measurements for population. Usually represented using Greek letters or capitalized English letters.
“N” for pop. Size; “” for pop mean
Sample Statistic: numerical measurements for sample. Usually represented using small English letters.
“n” for sample size; for sample mean
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4. Most commonly used -- Mean
Sample Mean:
“sample average”
Formula:
Population Mean:
“population average”
Characterize the center of the data distribution
The most commonly used data measure
Ways to compute the mean:
Use calculator.
Use Excel. (function: average)
Reading information from the chart.
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5. Sensitivity to outliers
Compute the mean for the following 2 groups of data
Sensitivity to outliers
If the mayor decide to provide more public facilities to poor
communities, and the decision is made based on whether
the mean income in the community is below $50,000 per year.
Does such a decision make sense?
Household income in community a: (Unit =10000$)
Household income in community b: (Unit =10000$) #1 #2 #3 #4 #5 #6 #7 #8 5 4 3 #1 #2 #3 #4 #5 #6 #7 #8 5 4 3
100
4.125 16
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6. Exercise:
The manager of a small hotel in Foster City, CA, was asked by the corporate VP to analyze the Sunday night registration information for the past eight weeks. Data on three variables were collected:
x1 = total number of rooms rented
x2 = total dollar revenue from the room rentals
x3 = number of customer complaints that came from guests each Sunday
Tasks:
Create a histogram for the distribution of number of customer complaints every day
Calculate the average number of rooms rented, the average revenue, the average number of complaints per day.
Calculate the average number of complaints per room rented
Explain the difference between “the average compliant per day” and “the average complaint per room rented“ from a managerial perspective.
Week
Rooms Rented
Revenue
Complaints 1 22
$1,870 2 13
$1,590 3 10
$1,760 4 16
$2,345 5 23
$4,563 6
$1,630 7 11
$2,156 8
$1,756
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7. Compute the mean from frequency table
Below is a frequency table showing the number of days the teams finish their projects
How many days on average does a team finish one project?
Create a histogram using the data on the left, locate the mean on the graph.
How to describe the shape of the histogram?
What is the relationship between the mean and peak?
Use relative frequency to find out the mean.
Days to Complete
Frequency 5 4
=5*4 6 12
=6*12 7 8
=7*8
=8*6 9
=9*4 10 2
=10*2
6.31 days
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8. Estimating the mean from Histogram
Treat Histogram as a frequency table, use the mid-value to estimate each range.
Mathematical Expression: if sample, if population
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9. BUS304 – Data Characterization
Weighted Mean
The mean assumes that each piece of information equally.
E.g. students’ GPA and score calculation.
Weights are subjective.
E.g. Different instructors assign different weights to homework and exams.
Frequency table can be considered as an example of weighted mean (higher weights when higher frequency)
Days to Complete
Frequency
Relative Frequency 5 4
11.11% 6 12
33.33% 7 8
22.22%
16.67% 9 10 2
5.56%
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10. BUS304 – Data Characterization
Exercise:
Estimate the mean based on the following histogram
There are 30 full time faculty in CoBA. Their average age was 43 in In 2008, one new faculty with age 30 was hired and one faculty retired at 65. What is the new mean age for CoBA faculty?
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11. BUS304 – Data Characterization
Variance
A measure of data spread.
Also called “the average of squared deviations from the mean”
The larger the variance, the fat the histogram
-- sample variance population variance
Note the difference!
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12. Steps to compute the variance
Identify whether the data are of a population or sample
(the formulae are different.)
Use the following table to compute the deviation:
Find out the mean:
Find out the distance
(fill out the 2nd column)
Find out the squared distance (the 3rd column)
Add up the 3rd column
divided by
population size; or
sample size -1
Data list
Distance from
the mean
Square
the distance 5 4 3 2
= =1.167
=(1.167)2=1.36
If the list of data is a population, what is the population variance?
If the list of data is a sample, what is the sample variance?
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13. Comparing variance vs. histogram
Find the variance for the following groups of sample data:
Compare the mean and variance.
Create the histogram to compare the distribution. 11 12 13 16 17 18 21 14 15 16 17 11 12 19 20
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14. What does variance mean?
Variance indicate variation:
The larger the variance, the more spread out the data.
Indicates unpredictability.
E.g.
Weather data: weather changes dramatically, hard to predict tomorrow’s temperature
(If look at temperature data: which has larger variance, Chicago or San Diego?)
Stock: more risk on returns.
A person’s performance: consistency. emotional…
Other examples?
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15. Use frequency table to compute the population variance:14 15 16 17
Data value
Frequency
Relative Frequency 14 1
0.125 15 3
0.375 16 17
Data
distance
square 14 15 16 17
Data
distance
square 14 15 16 17
Compute the
weighted average
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16. BUS304 – Data Characterization
Standard Deviation
Square root of variance.
An indicator of data deviation, can be directly compared to the mean.
Exercise: compute the standard deviation from the histogram on slide no. 5 and locate it on the histogram. OR
Sample variance
Population variance
Sample standard deviation
Population standard deviation
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17. BUS304 – Data Characterization
Empirical Rule
If the data is bell shaped (most of the time), then
68% of all data will fall in the range of
95% of all data will fall in the range of
99.7% of all data will fall in the range of
99.7%
95%
68%
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18. Other Numerical Measures
Median
Mode
Range
Percentiles
Quartiles, Interquartile range
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19. BUS304 – Data Characterization
-- The value which divides the data in half, with equal
sizes above and below
Median
The middle value
Steps:
Put your data in ordered array (sort)
If n (or N) is odd, the median is the middle number
(i.e. the th number)
If n (or N) is even, the median is the average of two middle numbers
(i.e. the average of the and the th numbers)
Reading information from the chart.
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20. Sensitivity to outliers
Median = 3
Median does not affected
by extreme values
Median = 2.5
4.125 16
Median = 3
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21. BUS304 – Data Characterization
Exercise 1 2 3 4 5 6 7 3 5 6 8 10 12 7 5 2 6 3 10 1 2 4 6 8 10 12 14 16 3 5 6 10 12 15 14 18 23 32 12 16 5 10 21 24 25 7
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22. BUS304 – Data Characterization
The value that occurs most often
Steps:
Put your data in ordered array (sort)
Find the data value(s) that repeats the most frequently
Mode
Mode does not affected
by extreme value either.
No Mode!
Mode=5
Reading information from the chart.
Boston Austin San Diego Los Angels
Mode=San Diego
Mode=5 and 9
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23. Find Mode and Median from Frequency Table
Below is a frequency table showing the number of days the teams finish their projects
Find the mean, median and mode.
Create a histogram, locate the mode, median and mode.
Describe the shape of the histogram, and find the relationship between mean, median and mode.
Days to Complete
Frequency 5 4 6 12 7 8 9 10 2
6.31 days
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24. Shape of a distribution
Mean = Median = Mode
Symmetric
Mean < Median < Mode
Left-Skewed
(Longer tail extends to left)
Mode < Median < Mean
Right-Skewed
(Longer tail extends to right)
Note that Mean is affected by the extreme value the most. So mean is always leaning towards the tail compared to the other two measures.
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25. Measures of center location
Mean
Median
Mode
Mean is generally used, unless extreme values (outliers) exist;
the next common is median, since the median is not sensitive to extreme values;
mode is sometime used when there is a really large frequency.
Think: Are house prices normally right-skewed or left-skewed? What measurement
People normally use to measure the house market?
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26. BUS304 – Data Characterization
Range
Simplest measure of variation
Describe how wide the data spread
Formula
Range = Maximum Value – Minimum Value
Example:
Range = = 13
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27. Range is affected the most by outliers.
Disadvantage of Range
Ignores the way in which data are distributed
Sensitive to outliers
1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = = 4
Range = = 5
1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = = 119
Range = = 5
Range is affected the most by outliers.
Feb 8, 2006
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28. BUS304 – Data Characterization
Other measures
Percentiles:
Measures the percentage of data below the value.
e.g. if the 60th percentile is 1240 (SAT score), that means there are 60% students getting a score less than 1240.
Correspondingly, there are 40% of students getting 1240 or higher.
How to find percentile? The pth percentile in an ordered array of n values is the value in the ith position, where
If the list of data is a population, what is the population variance?
If the list of data is a sample, what is the sample variance?
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29. BUS304 – Data Characterization
Example
Find the 80th percentile from the annual income data
Step:
Sort the data
Find the location for the 80th percentile:
Find the 80.8th person’s income
Where is the 80.8th person?
Combine the 80th and 81st numbers
80th  62245
81st  63485
80.8th  62245*20%+63485*80%=63237
1st th
80th 81st
80.8th should be
in between,
and closer to 81st.
80% because of the decimal is .8
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30. BUS304 – Data Characterization
Exercise
Find the 25th percentile
Find the 50th percentile
Find the 75th percentile
Explain the meaning of 50th percentile? Have you learnt a similar measurement?
How many people have income levels between the 25th and the 50th percentiles?
How many people have income levels between 50th and the 75th percentile?
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31. BUS304 – Data Characterization
Quartiles
The 25th, 50th, and 75th percentiles
Called the first, second, and third quartiles, respectively.
Written as Q1, Q2, Q3, respectively.
The quartiles split the ranked data into 4 equal groups.
25%
25%
25%
25% Q1 Q2 Q3
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32. BUS304 – Data Characterization
Example:
Example: Find the first quartile in the data sample:
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33. BUS304 – Data Characterization
Interquartile Range
Recall:
Range? Disadvantage of range?
Interquartile Range:
Interquartile Range = Q3 – Q1
Example:
Q1=13.5
Q3=19
Interquartile range = Q3 – Q1 = 19 – 13.5 = 5.5
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34. BUS304 – Data Characterization
Summary
Understand and compute the following two sets of data measures:
Measures of central tendency
Mean, Median, and Mode
Measures of variation
Range, Variance, and Standard deviation
Other ways to describe data:
Percentiles, Quartiles, Interquartile range
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