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Quantitative Methods
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QMS 230
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Chapter 3

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Quantitative Methods

QMS 230

Douglas Mc Kessock

Fall

Description

CQMS204- Statistics for Management Chapter#3
Chapter3 Numerical Descriptive
Measures
Measures of Central Tendency (use calc)
Outcomes:
1. Differentiate between the mean, the median, and the mode of data
2. State the advantages and disadvantages of the three measures
Central Values
1. The mean, the mode and the median are referred to as measures of
central tendency.
2. These three measures are indicators of typical middle values in the data
set.
3. The mean is the arithmetic average of all values in the data set. 1 answer
4. The median indicates the halfway point.
5. The mode is the most frequently occurring value. Multiple answers
The Mean
a. The Arithmetic Mean when all values = the same weight
sample mean
x
x = n
population mean
= x
N
b. The Weighted Mean when values are weighted different (ex. Marks)
wx
x =
w
where
x = the sample mean;
= the population mean;
x = the observed values;
n = the number of observations in the sample;
N =the number of observations in the population;
w = weights, i.e numbers to indicate the relative importance
of each data value in the overall result;
wx = the weighted values of the observed values;
wx = the sum of the weighted values
Note: the symol read sigma and denotes the summation of values.
Winter2011 Page1 CQMS204- Statistics for Management Chapter#3
c. The population mean of Grouped Data
fm
=
f
where
= the mean of the data set;
m
= the individual class midpoints; midpoint = lower + upper / 2
f = the number of observations in each class;
f = the total number of observations in the data set;
fm = the sum of the weighted values of the class midpoints
The Median
The median of Ungrouped Data
Position number of median in an array =+ 1
2
The Mode
the most frequently occurring value
Example1
The annual incomes of a sample of middle-management employees at Ryerson
Inc. are: $45,000, $50,000, $55,000, $50,000, $53,000, $47,000
a. Determine the sample mean, mode and median.
b. Is the mean we determined in (a) a statistic or a parameter? Why?
a) Mean (or x bar): 50,000
Mode: 50,000
Median: 50,000
b) Statistics. Because we are calculating for a sample.
Example2
There are 11 salespeople employed by Midtown Inc. The numbers of new
cars sold last month by the respective salespeople were:
15, 23, 28, 9, 19, 18, 10, 10, 8, 28, 19
a. Determine the population mean, mode and median.
b. Is the mean we determined in (a) a statistic or a parameter? Why?
a) Mean (or mew (because its population): 17
Mode: 10, 19 & 28
Median: 18
b) Parameter. Because it is a population.
Winter2011 Page2 CQMS204- Statistics for Management Chapter#3
10% rule:
1. Calculate the difference between the mean and median
2. Calculate 10% of the smaller of the mean or m~dian
3. If the difference <10% of smaller conclude x = x in which case the mean
is the preferred measure.
4. If the difference >10% of smaller conclude x x in which case the
median is the preferred measure.
Example3
a. The following data represents the prices of houses sold last week in a
particular neighbourhood.
$200,000 $205,000 $210,000 $220,000 $225,000
What is the appropriate measure of central tendency for these data?
Why?
1. Mean (x bar): 212000
Median (x with squiggly on top): 210000
Mean Median = 212,000 210,000 = 2,000
2. 210,000 (cause its smaller) * 0.1 = 21,000
3. Since 2,000 < 21,000, therefore you can use Mean or Median. (refer to 3
for 10% rule). However mean is better! So use mean.
b. The following data represents the prices of houses sold last week in a
particular neighbourhood.
$200,000 $205,000 $210,000 $220,000 $425,000
What is the appropriate measure of central tendency for these data?
Why?
1. Mean (x bar): 252,000
Median (x with squiggly on top): 210,000
Mean Median = 252,000 210,000 = 42,000
2. 210,000 * 0.1 = 21,000
3. 42,000 > 21,000, therefore it is Median. (refer to 4 for 10% rule)
Example4
A&A Marketing Research Company is studying the number of times the ATM
located in G&G Grocery Store at Yong and Finch is used per day. Following are
the numbers of times the machine was used over each of the last 30 days.
82, 79, 84, 78, 82, 82, 77, 83, 88, 85,
84, 77, 79, 78, 79, 75, 78, 82, 82, 83,
82, 74, 78, 78, 78, 82, 78, 83, 85, 83.
a. Determine the sample mean, mode and median.
b. Is the mean we determined in (a) a statistic or a parameter? Why?
Winter2011 Page3 CQMS204- Statistics for Management Chapter#3
a) Mean (x bar = sample): 80.6
Mode: 78, 82
Median: 82
b) Statistic because we are using sample.
Example5
The Bookstall Inc. is a specialty bookstore concentrating on used books sold via
the internet. Paperbacks are sold at $1.00 each, and hardcover books are $4.50.
Of the 60 books sold last Thursday morning, 48 were paperback and the rest
were hardcover. What was the weighted mean price of a book?
X W (frequency)
1 48
4.50 12
Weighted Mean (x bar) = $1.7
Example6
There is intense competition among the long distance phone business. In an
effort to study the actual phone usage of residential customers, an independent
agency gathered the following sample information on the number of phone calls
per household for the last week.
Number of phone calls 2 3 4 5 6 7
(X)
Number of 15 11 21 12 23 18
households(W)
Determine the sample mean, mode and median.
X W(frequency)
2 15
3 11
4 21
5 12
6 23
7 18
Mean (x bar): 4.71
Mode: 6 (because it occurs the most)
Median: 5
Example7
Winter2011 Page4

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