business and finance colleges principles of statistics eng. heba hamad week 6 - 2008
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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