Statistics lecture 2
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Statistics lecture 2. Summarizing and Displaying Measurement Data. Thought Question 1. If a study shows that daily use of a certain expensive exercise machine resulted in an average loss of 10 pounds, what more would you want to know about the numbers than just the average?.

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Statistics lecture 2

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Statistics lecture 2

Summarizing and Displaying

Measurement Data


Thought Question 1

If a study shows that daily use of a certain expensive exercise machine resulted in an average loss of 10 pounds, what more would you want to know about the numbers than just the average?


Thought Question 2

Imagine you wanted to compare the cost of living in two different cities. You get local papers and write down the rental costs of 50 apartments in each place. How would you summarize the values in order to compare the two places?


Goals for Lecture 2

  • Realize that summarizing important features of a list of numbers gives more information than just the unordered list.

  • Understand the concept of the shape of a set of numbers.

  • Learn how to make stemplots and histograms

  • Understand summary measures like the mean and standard deviation


Height in centimetersofDoig’s students

170, 163, 178, 163, 168, 165, 170, 155, 191, 178, 175, 185, 183, 165, 165, 180, 185, 165, 168, 152, 178, 183, 157, 165, 183, 157, 170, 168, 163, 165, 180, 163, 140, 163, 163, 163, 165, 178, 150, 170, 165, 165, 157, 165, 173, 160, 163, 165, 178, 173, 180, 196, 185, 175, 160, 168, 193, 173, 183, 165, 163, 175, 168, 160, 208, 157, 180, 170, 155, 173, 178, 170, 157, 163, 163, 180, 170, 165, 170, 170, 180, 168, 155, 175, 168, 147, 191, 178, 173, 170, 178, 185, 152, 170, 175, 178, 163, 175, 175, 165, 175, 175, 157, 163, 165, 160, 178, 152, 160, 170, 170, 160, 157,


Height in centimetersofDoig’s students (sorted)

208, 196, 193, 191, 191, 185, 185, 185, 185, 183, 183, 183, 183, 180, 180, 180, 180, 180, 180, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 175, 175, 175, 175, 175, 175, 175, 175, 175, 173, 173, 173, 173, 173, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 168, 168, 168, 168, 168, 168, 168, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 160, 160, 160, 160, 160, 160, 157, 157, 157, 157, 157, 157, 157, 155, 155, 155, 152, 152, 152, 150, 147, 140


Three Useful Featuresof a Set of Data

  • The Center

  • The Variability

  • The Shape


The Center

  • Mean (average): Total of the values, divided by the number of values

  • Median: The middle value of an ordered list of values

  • Mode: The most common value

  • Outliers: Atypical values far from the center


Example: 2006 Baseball Salaries

  • Average: $2,827,104

  • Median: $950,000

  • Mode: $327,000 (also the minimum)

  • Outlier: $21.7 million (Alex Rodriguez of the NY Yankees)


The Variability

Some measures of variability:

  • Maximum and minimum: Largest and smallest values

  • Range: The distance between the largest and smallest values

  • Quartiles: The medians of each half of the ordered list of values

  • Standard deviation: Think of it as the average distance of all the values from the mean.


What is “normal”?

  • Don’t consider the average to be “normal”

  • Variability is normal

  • Anything within about 3 standard deviations of the mean is “normal”


R

A

N

G

E

Measuring Variability

125 Highest

120

110 Upper quartile

110 Interquartile

100 Median Range

90

90 Lower quartile

80

75 Lowest


Standard Deviation:How to Compute

  • Data: 90, 90, 100, 110, 110

    • Mean: 100

    • Deviations from mean: -10, -10, 0, 10, 10

    • Devs squared: 100, 100, 0, 100, 100

    • Sum of squared devs: 400

    • Sum of sq devs/(n-1): 400/4=100 (variance)

    • Square root of variance: 10

      Therefore, the standard deviation is 10


Standard Deviation:How to Compute

  • Data: 50, 60, 100, 140, 150

    • Mean: 100

    • Deviations from mean: -50, -40, 0, 40, 50

    • Devs squared: 2500, 1600, 0, 2500, 1600

    • Sum of squared devs: 8200

    • Sum of sq devs/(n-1):8200/4=2050 (variance)

    • Square root of variance: 45.3

      Therefore, the standard deviation is 45.3


The Shape

The shape of a list of values will tell you important things about how the values are distributed.

To visualize the shape of a list of values, plot them using:

  • a stemplot (also called stem-and-leaf)

  • a histogram

  • or a smooth line (next lecture)


How to Make a Stemplot

  • Divide the range into equal units, so that the first few digits can be used as the stems. (Ideally, 6-15 stems.)

  • Attach a leaf, made of the next digit, to represent each data point. (Ignore any remaining digits.)


How to Make a Stemplot

Ages in years:

42.2, 22.7, 21.2, 65.4, 29.3, 22.3, 21.5, 20.7, 29.4, 23.1, 22.9, 21.5, 21.4, 21.3, 21.3, 21.2, 21.2, 21.1, 20.8, 30.2, 25.7, 24.5, 23.2, 22.3, 22.2, 22.2, 22.2, 22.1, 21.9, 21.8, 21.7, 21.7, 21.6, 21.4, 21.3, 21.2, 21.2, 21.2, 21.2, 21.2, 21.1, 21.1, 20.8, 20.7, 20.7, 20.1, 20.0, 19.5, 35.8, 26.1, 22.3, 22.2, 21.8, 21.5, 20.4, 47.5, 45.5, 30.6, 28.1, 27.4, 26.5, 24.1, 23.3, 23.3, 22.9, 22.9, 22.6, 22.4, 22.4, 22.3, 22.3, 22.0, 21.9, 21.9, 21.8, 21.7, 21.7, 21.7, 21.6, 21.6, 21.6, 21.5, 21.5, 21.5, 21.4, 21.2, 21.2, 21.2, 21.1, 21.1, 21.0, 20.9, 20.9, 20.8, 20.8, 20.8, 20.8, 20.8, 20.6, 20.6, 20.6, 20.5, 20.5, 20.5, 20.5, 20.4, 20.4, 20.3, 20.2, 19.9, 19.6, 63.2, 55.0


How to Make a Stemplot

19 |

20 |

21 |

22 |

23 |


How to Make a Stemplot

19 | 5

20 | 0123444

21 | 0111112222222222

22 | 01222

23 | 12


Stemplot of class ages up to 30


Another Age Stemplot(Each Stem = 5 Years)

2| (20-24)

2| (25-29)

3| (30-34)

3| (35-39)

4| (40-44)

4| (45-49)

5|(50-54)


Another Age Stemplot

2|000000000000001111111111111111111111111111111

11111111122222222222222222222222222223333333334

2|56677899

3|01

3|6

4|2

4|57

5|

5|5

6|3

6|5


Histogram

  • Shows the shape of a set of values, similar to a stemplot

  • More useful for large data sets because you don’t have to enter every value

  • X-axis: Range of possible values

  • Y-axis: The count of each possible value


Height in InchesofDoig’s Students


Shape: Symmetric Data Set


Shape: Right-Skewed Data Set

(15-19)


Shape: Left-Skewed Data Set

(15-19)


Unimodal or Bimodal?


Bimodal! But why?


Height by Gender


R

A

N

G

E

Measuring Variability

125 Highest

120

110 Upper quartile

110 Interquartile

100 Median Range

90

90 Lower quartile

80

75 Lowest


Five-Number Summary

Median

Lower quartile Upper quartile

Lowest value Highest value


Five-Number Summaryof MCO302 Height in Centimeters

  • Lowest140

  • First quartile163

  • Median168

  • Third quartile178

  • Highest208


Heights

  • Women:140, 150, 152, 152, 155, 155, 155, 157, 157, 157, 157, 157, 157, 157, 160, 160, 160, 160, 160, 160, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 163, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 165, 168, 168, 168, 168, 168, 168, 168, 168, 168, 168, 168, 170, 170, 170, 170, 170, 170, 170, 170, 170, 170, 173, 173, 173, 173, 175, 175, 175, 175, 175, 175, 178, 178, 180, 180, 180, 208

  • Men:147, 152, 163, 165, 168, 170, 170, 170, 173, 175, 175, 175, 178, 178, 178, 178, 178, 178, 178, 178, 180, 180, 180, 183, 183, 183, 183, 185, 185, 185, 185, 191, 191, 193, 196


Five-Number Summaryby Gender

WomenMen

Lowest 140147

First quartile163174

Median 165178

Third quartile170183

Highest 208196


Ages of Death

  • Presidents: 67, 90, 83, 85, 73, 80, 78, 79, 68, 71, 53, 65, 74, 64, 77, 56, 66, 63, 70, 49, 56, 71, 67, 71, 58, 60, 72, 67, 57, 60, 90, 63, 88, 78, 46, 64, 81, 93

  • Vice-Presidents: 90, 83, 80, 73, 70, 51, 68, 79, 70, 71, 72, 74, 67, 54, 81, 66, 62, 63, 68, 57, 66, 96, 78, 55, 60, 66, 57, 71, 60, 85, 76, 8, 77, 88, 78, 81, 64, 66, 70


Age of Death:Five-Number Summary

Presidents Vice-Presidents

Lowest age4651

Lower quartile6364

Median age6970

Upper quartile7879

Highest age9398


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