Business and Finance Colleges

Principles of Statistics

Eng. Heba Hamad

week 6 - 2008

Introduction to Statistics

The Weighted Mean andThe Weighted Mean andWorking with Grouped DataWorking with Grouped Data

Weighted MeanMean for Grouped DataVariance for Grouped DataStandard Deviation for Grouped Data

Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

Introduction to Statistics

Introduction to Statistics

Weighted Mean

i i

i

wxx

w

i i

i

wxx

w

where:where:

xxii = value of observation = value of observation ii

wwi i = weight for observation = weight for observation ii

Introduction to Statistics

GPA Example Grade # Credits (Weight) Product A 4 4 16 B 3 3 9 B 3 2 6 C 2 1 2

10 33 (sum of above)

A = 4, B = 3, C = 2

GPA = 33/10 = 3.3

Grouped Data The weighted mean computation can be used toThe weighted mean computation can be used to obtain approximations of the mean, variance, andobtain approximations of the mean, variance, and standard deviation for the grouped data.standard deviation for the grouped data. To compute the weighted mean, we treat theTo compute the weighted mean, we treat the midpoint of each classmidpoint of each class as though it were the mean as though it were the mean of all items in the class.of all items in the class. We compute a weighted mean of the class midpointsWe compute a weighted mean of the class midpoints using the using the class frequencies as weightsclass frequencies as weights.. Similarly, in computing the variance and standardSimilarly, in computing the variance and standard deviation, the class frequencies are used as weights.deviation, the class frequencies are used as weights.

Introduction to Statistics

Mean for Grouped Data

i if Mx

n i if M

xn

N

Mf iiN

Mf ii

where: where:

ffi i = frequency of class = frequency of class ii

MMi i = midpoint of class = midpoint of class ii

Sample Data

Population Data

Introduction to Statistics

Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the

form of a frequency distribution.

Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Sample Mean for Grouped DataSample Mean for Grouped Data

Sample Mean for Grouped DataSample Mean for Grouped Data

This approximationThis approximationdiffers by $2.41 fromdiffers by $2.41 fromthe actual samplethe actual samplemean of $490.80.mean of $490.80.

34,525 493.21

70x

34,525 493.21

70x

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

f iM i

3436.07641.55634.03916.03566.52118.01099.02278.01179.03657.034525.0

Variance for Grouped Data

sf M xn

i i22

1

( )s

f M xn

i i22

1

( )

22

f M

Ni i( ) 2

2

f M

Ni i( )

For sample data

For population data

Introduction to Statistics

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

Sample Variance for Grouped DataSample Variance for Grouped Data

M i - x

-63.7-43.7-23.7-3.716.336.356.376.396.3116.3

f i(M i - x )2

32471.7132479.596745.97110.11

1857.555267.866337.13

23280.6618543.5381140.18

208234.29

(M i - x )2

4058.961910.56562.1613.76

265.361316.963168.565820.169271.76

13523.36

continuedcontinued

Introduction to Statistics

3,017.89 54.94s 3,017.89 54.94s

ss22 = 208,234.29/(70 – 1) = 3,017.89 = 208,234.29/(70 – 1) = 3,017.89

This approximation differs by only $.20 This approximation differs by only $.20

from the actual standard deviation of $54.74.from the actual standard deviation of $54.74.

Sample Variance for Grouped DataSample Variance for Grouped Data

Sample Variance

Sample Standard Deviation

Introduction to Statistics

General Examples

Introduction to Statistics

Example 1

Fine mean, median, mode.

Introduction to Statistics

1612

192

n

xx

x27817111525161414141318

192

x81113141414151617182527

Solution

Introduction to Statistics

Example 2

0)5(6

9144.4)025.4(6

)1(

2

22

s

nn

xxns

Fine Standard deviation, variance for each of the two sample

x x2

0.8192 0.67110.815 0.66420.8163 0.66630.8211 0.67420.8181 0.66930.8247 0.68014.9144 4.025

Coke

Introduction to Statistics

Fine Standard deviation, variance for each of the two sample

x x2

0.8258 0.68190.8156 0.66520.8211 0.67420.817 0.66750.8216 0.6750.8302 0.68924.9313 4.053

Pepsi

32

22

1006.3)5(6

9313.4)053.4(6

)1(

s

nn

xxns

Introduction to Statistics

Example 3

xz

62.0

20.98

9.262.0

2.98100

100 )

z

xa

262.0

2.9896.96

96.96 )

z

xb

062.0

2.982.98

2.98 )

z

xc

Introduction to Statistics

Example 4Fine the indicated quartile or percentilea) Q1, b) Q3, c) P80, d) P33

Introduction to Statistics

8152.0 12 88.1136100

33

0.8229 92 8.2836100

80

th33

th80

P

P

Introduction to Statistics

Exercise

Fine mean, median, mode, midrange, range, standard deviation, variance, P30

Age of US President

Introduction to Statistics