Lauretta J. Fox
Statistics play a role in all our lives. From time to time we provide data or collect data to be organized, analyzed, and used in making decisions. For example, the registrar of vital statistics records our date of birth and our date of death. Periodically we are counted in the population census taken by the federal government. School children are always concerned about their scholastic averages. When planning vacations we are interested in data provided by the weather bureau. We try to predict the outcome of elections by analyzing the results of political polls that have been taken. Business people conduct surveys to find out what products are most frequently sought by consumers. Sports fans compare batting averages, wins and losses of various teams. Although our list could continue indefinitely, it is sufficient to say that we encounter statistics in many facets of our lives, and all school children should be introduced to basic statistical terms and concepts.
In this unit of study we will try to improve the students’ understanding of the elementary topics included in statistics. The unit will clearly define the arithmetic mean, the median, the mode and the range of a group of numbers. Methods for computing these measures of central tendency will be discussed. Frequency tables, histograms and frequency polygons will be explained and constructed. Following the explanation of each topic, a set of practice exercises will be included.
There are several basic objectives for this unit of study. Upon completion of the unit, the student will be able to:
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— understand and appreciate the use of statistics in everyday life.
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— define basic terms used in statistics.
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— compute simple measures of central tendency.
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— construct tables, bar graphs and line graphs that display these measures of central tendency.
The material developed here may be used at the following levels of instruction: (1) in seventh or eighth grade arithmetic classes; (2) in high school applied mathematics classes; (3) in high school consumer mathematics classes; (4) in adult basic education classes.
Arithmetic
Mean
The
arithmetic
mean
is another name for the average of a set of measures. To calculate the arithmetic mean, add the members of the set and divide the sum by the number of items in the set.
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Example 1
: Find the arithmetic mean of the following set of numbers: 15, 10, 12, 18, 20.
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Solution
: (15+ 10+ 12+18+20) $dv$ 5 = 75 $dv$ 5 = 15 The arithmetic mean is 15.
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Example
2
: On four English tests John received grades of 92,80, 88, 94. What grade must he receive on the next test if he wishes to maintain an average of 90?
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Solution
: If John wishes to have an average of 90 on five tests, the sum of his scores on the five tests must be 5 x 90 or 450. His total score on the first four tests is 92+80+88 + 94 or 354. He must receive 450 354 or 96 on the fifth test to maintain an average of 90 on the five tests.
Sometimes an item appears in a set of measures more than once. The number of times any item occurs in the set is its
frequency
. To find the arithmetic mean of a set of measures when some items occur several times, multiply each item in the set by its frequency and divide the sum of these products by the total number of items in the set.
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Example
: Find the arithmetic mean of the following numbers: 8, 6, 10, 11, 12, 6, 6, 8, 14, 15, 14, 14, 10, 12, 6, 11.
(figure available in print form)
Sum of Products:
24+ 16+ 20+ 22+24+ 42+30+20 = 198
Total Number of Items:
4+2+2+2+2+3+2+1 = 18
Sum of Products $dv$ Total Number of Items:
198 $dv$ 18 = 11
The arithmetic mean is 11.
Class
Assignment
:
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1.) Measure the height of each student in the class. Find the average height of the students in the class.
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2.) Keep a record of your test scores for one marking period. At the end of the marking period find the arithmetic mean of the test scores.
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Exercises Solve the following set of problems.
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1.) Five employees of Brad’s Department Store earned the following hourly wages: $4.35, $3.67, $3.36, $5.00 and $4.82. Find the average hourly rate of pay.
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2.) During a six week period Mary worked the following number of hours per week: 40, 42 1/2, 37 3/4, 48, 45, 44 3/4. Find the average number of hours that Mary worked per week.
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3.) Eight automobiles were priced at 10,499; $11,988; $7,444; $5,995; $14,999; $6,492; 10,750; and $7,937. What is the arithmetic mean of the prices?
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4.) Last winter a homeowner purchased 504 gallons of heating oil at an average cost of $1.23 per gallon. If he paid $1.19 per gallon for the first 354 gallons, what was the total cost of the remaining oil purchased?
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5.) Lou allows himself an average of $2.50 a day for lunch at work. If he spent $2.00 on Monday, $2.75 on Tuesday, $2.25 on Wednesday, and $2.50 on Thursday, how much may he spend for lunch on Friday?
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6.) Mrs. Smith purchased ten dozen rolls as follows: 2 dozen $1.19, 1 dozen $1.88, 3 dozen $.94, and 4 dozen $1.28. What average price per dozen did she pay for the ten dozen rolls?
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7.) On a vacation, Jack bought gasoline as follows: 10 gallons $1.29 per gallon, 15 gallons $1.19, and 12 gallons $1.23. What was the average price per gallon?
Median
When the elements of a set of numbers have been arranged in ascending order, the number located in the middle of the set is the
median
of the set. The median divides the set of data into two equal parts. To determine which element of a set is the middle number, use the following formula:
Middle Number (Total Number of Elements + 1). Ö $ 2
If the set contains an even number of elements, the median is the average of the two middle numbers.
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Example
1
: Seven students received the following scores on an examination: 100, 04, 62, 95, 73, 71, 88. Find the median score.
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Solution
: Arrange the scores in order from lowest to highest: 62, 71, 73, 84, 88, 95, 100.
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(7+ 1) $dv$ 2 = 0 $dv$ 2 = 4. The fourth number of the set is the middle number. The median of the set is 84.
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Example
2
: Eight workers received the following hourly wages: $5.50, $3.75, $6.00, $4.25, $8.50, $3.90, $7.85, $4.80. What is the median wage?
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Solution: Arrange the wages in ascending order: $3.75, $3.90, $4.25, $4.80, $5.50 , $6.00, $7.85, $8.50.
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(8+ 1) $dv$ 2 = 9 = 2 = 4.5 The two middle numbers of the set are the fourth and fifth numbers. $4.80 and $5.50.
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($4.80 + $5.50) $dv$ 2 = $10.30 $dv$ 2 = $5.15
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The median wage is $5.15.
Class
Assignment
:
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1.) Find the median height of the students in your class.
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2.) Keep a record of your test scores for one marking period. At the end of the marking period, find the median test score.
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Exercises
: Solve the following problems.
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1.) Find the median of the given data:
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____
a.) 18.2, 16.8, 13.3, 19.4, 17.6
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____
b.) 68, 62, 64, 60, 63, 61, 59
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____
c.) 237, 225, 230, 228, 236, 232
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____
d.) 70, 74, 71, 72, 80, 78, 75, 69
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____
e.) $18.42, $16.74, $19.88, $15.64, $24.38, $14.76
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2.) Write mean or median to complete the sentence for the following data.
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____
a.) 5, 6, 2, 9, 8. The
is 6.
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____
b.) 11, 2, 3, 4, 6, 10. The
is 6.
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____
c.) 9.7, 4.2, 6.3, 8.5, 7.4. The
is 7.4.
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____
d.) 85, 83, 86, 90, 88, 91, 93. The is 88.
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____
e.) $3.60, $2.85, $4.90, $4.50, $5.00, $2.75. The is $4.05.
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3.) Are the following statements true or false?
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____
a.) The median of 12, 5, 6, 13, 10, 9, 7, 8, 14 is 10.
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____
b.) The median of 6.5, 4.0, 8.5, 2.5, 5.0, 3.5 is 4.5.
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____
c.) The mean of 2.8, 7.6, 5.4, 8.2, 3.9 is 5.4.
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____
d.) The mean of 12.6, 6.8, 5.9, 10.7 is 9.
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____
e.) The median of 184, 200, 150, 148, 178 is 178.
Mode
and
Range
The
mode
of a set of numbers is the element that appears most frequently in the set. There can be more than one mode in a set of numbers. A set that has two modes is
bimodal
. One that has three modes is
trimodal
. If no element of a set appears more often than any other element, the set has no mode. The mode is an important measure for business people. It tells them what items are most popular with consumers.
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Example
1
: Find the mode of the following set of numbers: 2, 5, 4, 2, 3, 7, 9, 5, 2, 4, 8.
|
Solution:
|
Element
|
Frequency
|
(figure available in print form)
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The number 2 occurs most frequently, hence 2 is the mode of the set.
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Example 2
: Find the mode of the following set of numbers: 10, 11, 15, 10, 9, 11, 10, 12, 11, 12.
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Solution
:
(figure available in print form)
-
The numbers 10 and 11 each appear three times. The set has two modes: 10 and 11.
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Example
3
: Find the mode of the following set of numbers: 1, 2, 3, 4, 5.
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Solution
: No number appears more than any other one in the set. The set has no mode.
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The
range
of a set of numbers is the difference between the highest and lowest numbers of the set. To find the range of a set of numbers, use the following formula:
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Range = Highest Number Lowest Number
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Example
: What is the range of the following set of numbers? 5, 8, 10, 15, 7, 6, 20, 9
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Solution
: The highest number of the set is 20. The lowest number of the set is 5.
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20 5 = 15. The range of the set is 15.
Class
Assignment
:
-
1.) Find the range in height of the students in your class.
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2.) Find the range of test scores obtained by students in the class for one marking period.
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Exercises
: Solve the following problems.
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1.) Find the mode of the given data.
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____
a.) 3, 5, 6, 7, 6, 2.
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____
b.) 14, 15, 14, 17, 15, 15, 18, 19.
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____
c.) 231, 237, 248, 244.
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____
d.) 84, 86, 87, 84, 86, 89, 90, 87, 87, 84, 86.
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____
e.) 29, 30, 31, 28, 29, 35.
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2.) Find the range of the given data.
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____
a.) 3.6, 9.2, 5.8, 7.4, 12.1.
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____
b.) $22.54, $19.82, $50.00, $35.60, $42.78, $15.63.
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____
c.) 70 64, 98, 69, 82, 85, 59.
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____
d.) 12, 6 1/2, 13, 8 1/2, 11, 15 1/4.
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____
e.) 26.3, 9.27, 15.7, 28.9, 18.8.
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3.) Write mean, mode, median or range to complete the sentence for the given data.
-
____
a.) 17, 19, 17, 15, 20, 16, 18. The
is 17.
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____
b.) 78, 68, 82, 96, 84, 90, 76. The
is 82.
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____
c.) 42, 69, 53, 75, 97, 88, 38. The
is 69.
-
____
d.) 22, 10, 42, 39, 27, 32, 49. The
is 39.
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4.) Complete the table for the given data.
(figure available in print form)
Frequency
Table
Large amounts of information can easily be organized, read and understood by listing data in a frequency table. A
frequency
table
is a chart in which the members of a set are tallied, and the total count for each item is recorded. If the set has a wide range of elements, it may be divided into equal intervals to make the frequency table shorter.
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Example
1
: The frequency table below shows data about favorite types of music. Use the table to answer the following questions.:
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____
a.) How many people are included in the survey?
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____
b.) What percent of the people surveyed preferred rock and roll music?
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____
c.) What is the ratio of people who prefer waltz music to those who prefer country and western music?
-
____
d.) If the number of people who prefer waltz music were increased by 6, What would the percent of increase be?
(figure available in print form)
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Solution
:
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a.) 17+ 14 + 18 +11 + 12 = 72 72 people are included in the survey.
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b.) 18 $dv$ 72 = .25 = 25% 25% of the people surveyed preferred rock and roll music.
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c.) 12:14 = 6:7. The ratio of people who prefer waltz music to those who prefer country and western music is 6:7.
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d.) 6/12 = x/100 12x = 600 x = 50
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The percent of increase is 50%.
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Example
2
: The prices of 25 different television sets are listed below. Show the data in a frequency table. Determine the median price and the range of prices.
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