1. Descriptive Statistics:
Part II
Each slide has its own narration in an audio file. For the explanation of any slide click on the audio icon to start it.
Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

2. Shape
A third important property of data – after location and dispersion - is its shape.
Shape can be described by degree of asymmetry (i.e., skewness).
mean > median positive or right-skewness
mean = median symmetric or zero-skewness
mean < median negative or left-skewness
Positive skewness can arise when the mean is increased by some unusually high values.
Negative skewness can arise when the mean is decreased by some unusually low values.
Descriptive Statistics II

3. Skewness Left skewed: Right skewed: Symmetric:
Source: Levine et al., Business Statistics, Pearson, 2013.
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4. Example: # hours to complete a task
This guy took a VERY long time!
Data (for n=12 employees): ┋ ┋ ┋ 𝑋 = 180/12 = 15 hours Median = 10 hours The (extremely slow) employee who took 63 hours to complete the task skewed the entire distributon to the right. s2 = 2868 / 11 = s = hours CV = 107.7%
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5. Example Using MS Excel
Scores of 17 students on a national calculus exam. Data:
0, 0, 10, 12, 15, 18, 20, 25, 30, 33, 34, 41, 56, 87, 92, 94, 95
Open MS Excel.
Go to Data Analysis—Analysis Tools — Descriptive Statistics.
If you do not have Data Analysis-Analysis Tools, you have to use the Add-in feature and add it to MS Excel.
Make sure to check the Summary Statistics box once you are in descriptive statistics.
See MS Excel Output on next slide.
Descriptive Statistics II

6. Using MS Excel From the output: mean is 38.94 median is 30 mode is 0
MS Excel uses a formula – the Pearson Coefficient of Skewness – to calculate skewness. You do not have to know the formula. If the coefficient is 0 or very close to it, you have a symmetric distribution.
From the output:
mean is 38.94
median is 30
mode is 0
standard deviation is 33.44
variance is
skewness is .78 (positive)
range is 95
n is 17
Descriptive Statistics II

7. Standardizing Data: Z-Scores
We can convert the original scores to new scores with 𝑋 = 0 and s = 1.
This will give us a pure number with no units of measurement.
Any score below the mean will now be negative.
Any score at the mean will be 0.
Any score above the mean will be positive.
Descriptive Statistics II

8. Standardizing Data: Z-Scores
To compute the Z-scores: 𝑍= 𝑋− 𝑋 𝑠 Example. Data: 0, 2, 4, 6, 8, 10 𝑋 = 30/6 = 5; s = 3.74 X  Z
0−5 3.74
-1.34 2
2−5 3.74
-.80 4
4−5 3.74
-.27 6
6−5 3.74
.27 8
8−5 3.74
.80 10
10−5 3.74
1.34
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9. Standardizing Data: Z-Scores
Data: Exam Scores
Original data
Change 7 to 97
Change 23 to 93 X Z 65
-0.45
-0.81
-1.40 73
-0.11
-0.38
-0.79 78
0.10
-0.10
-0.40 69
-0.28
-0.60
-1.09 7
-2.89
<= 97
0.94
1.07 23
-2.21
-3.12 93
0.76 98
0.99
1.14 99
1.05
1.22
0.90 75
-0.02
-0.27
-0.63 79
0.14
-0.05
-0.32 85
0.40
0.28 63
-0.53
-0.92
-1.56 67
-0.36
-0.70
-1.25 72
-0.15
-0.43
-0.86
0.73
0.72 95
0.82
0.83
0.91
Mean
75.57
79.86
83.19 s
23.75
18.24 s.
12.96
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10. Standardizing Data: Z-Scores
No matter what you are measuring, a Z-score of more than +5 or less than – 5 would indicate a very, very unusual score.
For standardized data, if it is normally distributed, 95% of the data will be between ±2 standard deviations about the mean.
If the data follows a normal distribution,
95% of the data will be between and
99.7% of the data will fall between -3 and +3.
99.99% of the data will fall between -4 and +4.
Worst case scenario: 75% of the data are between 2 standard deviations about the mean.
[Chebychev.]
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11. Smallest| Q1 | Median | Q3 | Largest
Five Number Summary
When examining a distribution for shape, sometime the five number summary is useful:
Smallest| Q1 | Median | Q3 | Largest
Example:
𝑋 = 15
5-number summary: 2 | 8 | 10 | | 63
This data is right-skewed.
In right-skewed distributions, the distance from Q3 to Xlargest (16.5 to 63) is significantly greater than the distance from Xsmallest to Q1(2 to 8). 2 3 8 9 10 12 15 18 22 63
Median Q1
Smallest Q3
Largest
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12. Boxplot
The boxplot is a way to graphically portray a distribution of data by means of its five-number summary.
Boxplot can be drawn along the horizontal or vertically.
Vertical line drawn within the box is the median
Vertical line at the left side of box is Q1
Vertical line at the right side of box is Q3
Line on left connects left side of box with Xsmallest (lower 25% of data)
Line on right connects right side of box with Xlargest (upper 25% of data)
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13. Boxplot
A “bell-shaped” symmetric data distribution would look like this:
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14. Categorical Data
We summarize categorical data using frequencies and graphical methods.
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15. Working with Frequencies
A frequency distribution records data grouped into classes and the number of observations that fell into each class.
A frequency distribution can be used for:
categorical data
numerical data that can be grouped into intervals
numerical data with repeated observations
A percentage distribution records the percent of the observations that fell into each class.
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16. Working with Frequencies
Example. A sample was taken of 200 professors at a (fictitious) local college. Each was asked for his or her (take-home) weekly salary. The responses ranged from about$520 to $590. If we wanted to display the data in, say, 7 equal intervals, we would use an interval width of $10.
Width of interval = 𝑅𝑎𝑛𝑔𝑒 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 = $70 7 = $10/class.
The Frequency / Percentage
Distribution: .
Take-home pay
frequency
percentage
520 and under
530 6 3 %
530 " "
540 30 15
540 " "
550 38 19
550 " "
560 52 26
560 " "
570 42 21
570 " "
580 24 12 to
590 8 4
200
100
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17. Working with Frequencies
A Cumulative Distribution focuses on the number or percentage of cases that lie below or above specified values rather than within intervals.
Take-home pay
frequency
percentage
less than
520
" "
530 6 3
540 36 18
550 74 37
560
126 63
570
168 84
580
192 96
590
200
100
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18. Working with Frequencies
The Frequency Histogram:
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19. Working with Frequencies
The Frequency Polygon
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20. Working with Frequencies
The Cumulative Frequency Distribution
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21. Descriptive Statistics – 2 variables
Categorical Data – graphical representation
Contingency Table
Side-by-Side Bar Chart
Numerical Data – looking for relationships in bivariate data
Scatter Plot
Correlation
The Regression Line
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22. The Contingency Table
Two categorical variables are most easily displayed in a contingency table. This is a table of two-way frequencies.
Example: “Who would you vote for in the next election?”
This also works for two-way percentages: .
Male
Female
Republican Candidate
250
500
Democrat Candidate
150
350
400
600
1000
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23. The Side-by-Side Bar Chart
Chart: Relative Performance (Source: Microsoft.com)
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24. The Scatter Plot
What can we do with 2 numerical variables? We can graph them.
Example – Grade and Height (in inches)
Y (Grade)
100 95 90 80 70 65 60 40 30 20
X (Height) 73 79 62 69 74 77 81 63 68
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25. The Scatter Plot
Correlation coefficient is r = .12
Coefficient of determination is r2 = .01
We will learn about the above measures, as well as more about scatter plots, in the topic onCORRELATION.
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26. Homework Practice, practice, practice.
As always, do lots and lots of problems. You can find these in the online lecture notes and homework assignments.
Descriptive Statistics II