Summary of Statistics II

8/12/2019 Summary of Statistics II

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Numerical

Summaries

of Data

Numerical Summary(primarily for quantitative variables)

Location Variation

Measures of Location

Give middle or typical valuesor central tendency.

Measures of Variation

Describe spread or scatter

or dispersion in the data.

Mode


1. Mean

2. Median

3. Mode


1. Mean

the center of gravity

of the data (histogram).

Population Mean =Sample Mean = X


1. Mean

the center of gravity

of the data (histogram).

Population Mean =Sample Mean = X


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formula for mean

Sample

Mean =Sum of observations

divided by

sample size

SXinX =

X1 + X2 + +Xnn=

Median- midpoint of distribution

At least half of the observations are

less than or equal to the median,

and at least half are

greater than or equal to the median.

Note: For n observations,

the median is located at the

n+ 1

2

in the ordered sample.

-th observation

Example 1

Data: 14, 18, 20, 12, 24, 15, 14

(n = 7odd)

Step 1: Order the data:

12, 14, 14, 15, 18, 20, 24

7 + 1

2= 4thlocation of median

q Median is the middle value.

q At least half the values are at or greater;

at least half are at or lower.

median example

Data: 14, 18, 20, 12, 24, 15, 14

(n = 7odd)

Step 1: Order the data:

12, 14, 14, 15, 18, 20, 24

94 (outlier)

94

Original: X = 16.71

with outlier: X = 26.71

Example 2

q Median is stillthe middle value.q Median isresistant to outliers.

Data: 14, 18, 20, 12, 24, 15, 14, 214

(n = 8even, outlier)

1st: Order the data:

12, 14, 14, 15, 18, 20, 24, 214

q Median is the average of the two middle values.

q Exactly half the values are greater, half lower.

Example 3

8 + 1

2= 4.5thlocation of median

16.5


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1. Order the data.

2. For odd n, the median isthe center observation.

3. For even n, the median is

the average of the two center

observations.

Summary for Finding Median 3. Mode - most frequentlyoccurring value

In a histogram, modal class

is the one havinglargest frequency,

i.e., highest bar.

Good for a discrete quant variable (few values)

or a categorical variable.

What typeof variable is it?

q Ifcategorical, use the mode.

Average is meaningless;

look at percentages of occurrences.

q If variable is quantitative,

first look at a graph:

l Skewed or outliers?

l More or less symmetric?

Use median.

Use mean.

Numerical Summary

Location Variation

Mean

Median

ModePercentilesQuartiles

Range

Std. Deviation

IQR

Mutual Fund Selection

Two mutual funds; annual returns

for last three years from each.

Which fund would you choose?

Fund A: X = 10.0% Fund B: X = 10.0%8.0

12.0

10.0

60.0

-20.0

-10.0

Why does variation matter?Measures of Variation

1. Range

2. Variance &

Standard Deviation

3. Interquartile Range (IQR)


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Highest minus lowest

value in the sample.

1. Range Example 4: 3, 4, 1, 7, 4, 5

1 2 3 4 5 6 7

Example 5: 1, 1, 1, 7, 7, 7

1 2 3 4 5 6 7

Range = Hi - Lo = 7-1 = 6

Range = Hi - Lo = 7-1 = 6

How far are the data

from the middle,

on average?

2. Variance &

Standard Deviation

Sample Variance = s2Sample Std. Dev. = s

Population Variance = s2Population Std. Dev. = s

Notation:

Example 4: 3, 4, 1, 7, 4, 5 (miles)

1 2 3 4 5 6 7

X = 4.0- 3

- 1+1

+3

Avoid this by using either

1. absolute valueor

2. squaring

of the differences.

Note:

q The average of the deviations

from the mean will always be zero.We cannot let the negatives

cancel out the positives.

s

2

= n- 1

S(Xi-X)2

20

6-1=

= 4.0 miles2

Example 4 data:

Equation for Variance (for a sample):

3

4

1

7

4

5

x x - x (x x)2

-1

0

-3

3

0

1

24

x = 4.0Total 0

1

0

9

9

0

1

20

s2

Standard deviation:

s= s2

= 2.0 mi.


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Equation for Variance:

s2

= N

S(Xim)2

For a population:

s2=n- 1

S(XiX)2

For a sample:

Advantages: Good properties;

uses all the data.Disadvantages:

Units are squared.

Not resistant.

Variance

Standard Deviation

S= S2 The square root

of the variance.

= 4.0

= 2.0miles

Advantage:

Easier to interpret

than variance,

Units same as data.

If x= the 100pthpercentile, thenat least 100p% of data is x,at least 100(1-p)% of data is x.

Sample 100pthpercentile:

82% of the sample have scores 47,AND 18% have scores 47.

Example: You are told you scored 47;

then you hear 47 is at the 82ndpercentile.

1. Minimum2. 1stQuartile, Q1 = 25th ptile

3. Median

4. 3rdQuartile, Q3 = 75th ptile

5. Maximum

Five Number Summary

1st Quartile (25th percentile) :25% of the data values

lie at orbelowit.

3rd Quartile (75th percentile) :75%of the data values

lie at or belowit.

Quartiles:


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Quartiles

Q1: 25% of the data set is below the first quartile

Q2: 50% of the data set is below the second quartile

Q3: 75% of the data set is below the third quartile

25% 25% 25% 25%

Q3Q2Q1

Method 1: Percentile method

Q1located at position(n+1)*1/4



n Q1 Q2 Q3

5

8

11

31.5 4.5

4.52.25 6.75

63 9

median of observations

below theposition of

the median.

Q3 = median ofobservationsabove theposition of

the median.

Q1 =

Method 2: Median method Ordered data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Max =

Q3 =

Median =

Q1 =Min =

27.0

19.0

12.0

Q1= 15.0 Q3= 23.5

23.5

15.0

IQR = 8.5

Example 6

IQR = Q - Q13

q IQR is the rangeof the

middle50%of the data.

q Observations more than

1.5 IQRsbeyondquartiles

are considered outliers.

3. Interquartile Range (IQR)Which summary statistics

should I use?

Symmetric:

Use mean,

& std. dev.

Skewed right:

Use median,

& IQR.


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Boxplot

A graphically display of

the five number summary

(also called a box-and-whiskers plot)

Ordered data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Max =

Q3 =

Median =

Q1 =Min =

27.0

19.0

12.0

Q1= 15.0 Q3= 23.5

23.5

15.0

IQR = 8.5

Example 6


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28

22

24

12

14

16

18

20

26

Median

1

3Q

Q

Note:

Middle 50%of data are

within the box

Minimum

Maximum

Max =

Q3 =

Median =

Q1 =Min =

27.0

19.0

12.0

23.5

15.0

IQR = 8.5

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Summary of Statistics II