lecture 03. numerical descriptive statistics

7/28/2019 Lecture 03. Numerical Descriptive Statistics

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Statistics

ST 361: Statistics for Engineers

Numerical Descriptive Statistics

Kimberly Weems

[email protected]

5260 SAS Hall
mailto:[email protected]:[email protected]


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Numeric Measures

Why?Kim is in an introductory history class. On the

midterm exam Kim scored 64 out of 100? Did she

do well?

The class average was a 42.

By knowing the average for the class we can

make a comparison.


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Numeric Measures

Allow us to make comparisonsOf individuals to the group

Of group to other groups

Measures of centerGive an idea about the main chunk of the data


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Measures of Central Tendency

Mean-average Notation:

Population mean: mu

Sample mean: y-bary


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Summation Notation

1 2 3

1

...

n

i n

i

y y y y y


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Summation Notation

1 2 3

1

...

n

i n

i

y y y y y

Sum of y


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Summation Notation

1 2 3

1

...

n

i n

i

y y y y y

Sum of yIndividualvalues


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iyy

n


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iyy

n

Sum of thevalues

Sample size


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Median- Middle value in a data set whenvalues are put in increasing order

50% of values above and 50% below

If even number of observations just averagemiddle two.


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Simple Example:

A health researcher examined the amount ofsoda that a group of teenagers consumed

during a day. The resulting amounts in ounces

were: 9, 9, 6, 15, 12, 14, and 40.


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Simple Example:



were: 9, 9, 6, 15, 12, 14, and 40.

Mean: 15

9+9+6+15+12+14+40

7

10515

7

iyy

n


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Simple Example:

Soda consumed Median:

In increasing order 6 9 9 12 14 15 40


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Simple Example:

Soda consumed Median: 12

In increasing order 6 9 9 12 14 15 40


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Mean vs Median



were: 9, 9, 6, 15, 12, 14, and 40.

0 10 20 30 40


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Mean vs Median



were: 9, 9, 6, 15, 12, 14, and 40.

0 10 20 30 40

mean


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Mean vs Median



were: 9, 9, 6, 15, 12, 14, and 40.

0 10 20 30 40

meanmedian


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Problem with the mean:

Sensitive to unusual values and skewed datapulled away from the median

Skewed Right

Mean greater than median Skewed left

Mean less than median.

SymmetricMean and median are about the same.


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Trimmed Mean

A compromise between the average and themedian.

Less sensitive to outliers.

Observations are ordered from smallest to largest.A trimming percentage 100r% is chosen where r is

a number between 0 and 0.5.

Suppose r=0.1, so that the trimming percentage is

10%. Then if n=20, 10% of 20 is 2: the trimmedmean results from deleting (trimming) the largest

2 observations and the 2 smallest.


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CoalEmissions Uncertainty Project

(2009-10), Alissa Anderson,Colin Geisenhoffer, Brody

Heffner, Michael Shaw & Emily

Wisner

After 2% Trim

Before 2% Trim


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Measures of Variability

Why? Tell us about consistency and predictability

Allow comparison of groups

Gives scale of reference to compare individuals


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Range-difference in maximum and minimumHow spread out are the values

Soda Amounts: Range = 40-6=34

0 10 20 30 40


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Problem: Range only looks at two values.Does not quantify spread of the others.

Solution: Look at all values => How far are

they from mean Variance- summarizes distance between all

individuals and the mean


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Important notation:Population variance: 2sigma squared

Sample variance: s2


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Important Formula:

2

2 1

N

i

i

y

N


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Important Formula:

2

2 1

N

i

i

y

N

Squaredto get ridofnegatives

Calculateaverage ofthe

squareddistances


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Sample Variance

2

2 1

1

n

i

i

y ys

n


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Sample Variance

2

2 1

1

n

i

i

y ys

n

Divides byn-1 insteadof N


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Sample Variance

2

2 1

1

n

i

i

y ys

n

Divides byn-1 insteadof N

Sum ofsquares


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Simple Example:



were: 9, 9, 6, 15, 12, 14, and 40.


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What does it tell us?

By itself not much. Some people try lots of tricks to try to recreate

the data set from this number.

The purpose of the number is to make a

comparison with other data sets.

Example: Another group of teens had soda

consumption that had a variance of 473.2.

Other group was more spread out than our group.


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The Standard Deviation

Variance is not on the same scale as theoriginal data.

Standard Deviationsquare root of the

variance.Has the same units as original data

Allows more direct comparisons


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The Standard Deviation

For amount of soda

2

131.33 11.46

s s

s


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What does it tell us?

Understand variability in the data.Which is more consistent.


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City Temperature

City Raleigh

Mean 59

Median 61

SD 15


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City Temperature

City Raleigh Fargo

Mean 59 42

Median 61 43

SD 15 24


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City Temperature

City Raleigh Fargo Fairbanks

Mean 59 42

Median 61 43

SD 15 24


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City Temperature

City Raleigh Fargo Fairbanks

Mean 59 42 28

Median 61 43 31

SD 15 24 28


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City Temperature

City Raleigh Fargo Fairbanks Honolulu

Mean 59 42 28 77

Median 61 43 31 77

SD 15 24 28 3


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Coefficient of Variation

Coefficient of Variation (CV)ratio of standarddeviation to mean

Used to compare variability when scales are very

different.

. .s

C Vy


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Example:

Students in a midwestern state take a end ofgrade exam that has a maximum of 100 points.

A class testing a new teaching method had a

standard deviation of 10.

Students in an east coast state take an end of

grade exam that has a maximum of 500 points.

A class testing the new teaching method had a

standard deviation of 30. Which is more

varied?


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Example

The mean for the midwestern state was 70.

The mean for the east coast state was 350.

10. . 0.143 14.3%

70

sC V

y

30. . 0.086 8.6%

350

s

C V y

lecture 03. numerical descriptive statistics

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