Contents of Curriculum Unit 86.05.03:

Introduction

In this unit of study we will try to improve the students’ understanding of the elementary topics included in statistics. The unit will begin by discussing terms that are commonly used in statistics. It will then proceed to explain and construct frequency distributions, dot diagrams, histograms, frequency polygons and cumulative frequency polygons. Next, the unit will define and compute measures of central tendency including the mean, median and mode of a set of numbers. Measures of dispersion including range and standard deviation will be discussed. 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:

—define basic terms used in statistics.

—compute simple measures of central tendency.

—compute measures of dispersion.

—construct tables and graphs that display measures of central tendency.

The material developed here may be used as a whole unit, or parts of it may be extracted and taught in various courses. The elementary concepts may be incorporated into general mathematics classes in grades seven to twelve, and the more difficult parts may be used in advanced algebra classes in the high school. Depending upon the amount of material used, several days or several weeks may be allotted to teach the unit.

Definitions of Statistical Terms

Throughout the study of statistics certain basic terms occur frequently. Some of the more commonly used terms are defined below:

A population is a complete set of items that is being studied. It includes all members of the set. The set may refer to people, objects or measurements that have a common characteristic. Examples of a population are all high school students, all cats, all scholastic aptitude test scores.

A relatively small group of items selected from a population is a sample. If every member of the population has an equal chance of being selected for the sample, it is called a random sample. Examples of a sample are all algebra students at Central High School, or all Siamese cats.

Data are numbers or measurements that are collected. Data may include numbers of individuals that make up the census of a city, ages of pupils in a certain class, temperatures in a town during a given period of time, sales made by a company, or test scores made by ninth graders on a standardized test.

Variables are characteristics or attributes that enable us to distinguish one individual from another. They take on different values when different individuals are observed. Some variables are height, weight, age and price. Variables are the opposite of constants whose values never change.

Exercises:

1.) Tell whether each of the following is a variable or a constant:

a.) Scores obtained on a final examination by members of a statistics class.

b.) The cost of clothing purchased each year by secretaries.

c.) The number of days in the month of June.

d.) The time it takes to do grocery shopping.

e.) The age at which one may become a voter in the United States of America.

2.) Fill in the missing word to make a true statement.

a.) ____ are measurements obtained by observation.

b.) A ____ is a complete set of items.

c.) ________takes data collected from a small group and makes predictions about a wider sample.

d.) When every member of a set has an equal chance of being selected as part of a sample, the sample is called a ____ ____.

e.) Characteristics that vary from one individual to another are ____.

f.) The study that deals with methods of collecting, organizing and analyzing data is ____ ____.

Frequency Distributions

Example 1:

a.) How many people are included in the sample?

b.) What percent of the people surveyed preferred Chevrolets?

c.) What is the ratio of people who prefer Oldsmobiles to those who prefer Buicks?

d.) If the number of Subarus were increased by three, what would the percent of increase be?

Solution:

a.) 25 + 18 + 15 + 12 + 10 = 80 80 people are included in the survey.

b.) 25 Ö 80 = .3125 = 31.25% 31.25% of the people surveyed preferred Chevrolets.

c.) 15:10 = 3:2 The ratio of people who prefer Oldsmobiles to those who prefer Buicks is 3:2.

d.) 3:18=x:100 1:6 =x:100 6x=100 x =16 2/3 The percent of increase is 16 2/3%.

The following steps may be used to set up a frequency distribution:

1.) Select an appropriate number of intervals for the given data.

2.) Find the difference between the highest and lowest measurements in the data. Add one to the result end divide the sum by the number of intervals. If the quotient is not an integer, round it to the nearest odd integer. This will be the size or width of each interval and will be designated by the symbol w.

3.) The lowest number in the bottom interval will be the lowest measurement in the given data. Add (w-1) to this measurement to obtain the highest number in the bottom interval. The next interval begins at the integer following the highest number in the bottom group. Continue in this manner for each successive higher interval until every measurement has been placed in its proper group.

4.) After the intervals have been established, a tally mark is placed by the interval for each measurement in the group. The frequency, or number of measurements in each interval, is indicated with a numeral.

Example 2:

86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

Solution:

Highest Score—Lowest Score = 98Ð56 = 42 (42 + 1) = 10 = 43 $dv$ 10 = 4.3 Round to 5. The size of each interval is 5.

Scores	Tally	Frequency
96Ð100	111	3
91Ð95	1111	4
86Ð90	111	8
81Ð85	11	7
76Ð80	1	6
71Ð75	1111	5
66Ð70	111	3
61Ð65	11	2
56 Ð60	11	2

Although it is not necessary, it is often helpful for use in further analysis to have additional information in a frequency distribution. This additional information may include the midpoint of each interval, the percentage of the numbers in the frequency column relative to the total frequencies, the cumulative frequency of successive summation of entries in the frequency column, and the percentage of the cumulative frequency.

Example 3:

Solution:

(a) Since the width of each interval is 5, the third score is the midpoint of the interval. For example, the lowest interval contains the scores 56, 57, 58, 59, 60. 58 is the midpoint of this interval.

(b) To find the percentage of each frequency divide the frequency by the total number of measurements and change the resulting decimal to a percent. The frequency of the lowest interval is 2. The total number of measurements is 40. 2 $dv$ 40 = .05 = 5%

(c) The cumulative frequency at any interval may be obtained by successively adding the frequencies of all the groups from the lowest interval up to and including the given interval. The cumulative frequency of the interval 76-80 is 2+ 2+ 3+ 5+ 6 =18.

(d) To obtain the percentage of cumulative frequency relative to the total of the frequencies, divide the cumulative frequency by the total number of measurements. Change the resulting decimal to a percent. The percentage of the cumulative frequency in the interval 76-80 is 18 $dv$ 40 = .45= 45%. This figure may also be found by adding the percentage of frequency of all groups from the lowest up to and including the given interval.

Exercises:

1.) Ask the students in each of your classes which of the following colors they prefer—red, blue, yellow, green, brown, or purple. Construct a frequency distribution to display the results of your survey

a.) How many people are included in the sample?

c.) What is the ratio of people who prefer green to those who prefer blue?

d.) What is the most popular color?

e.) What is the least popular color?

f.) If the number of people who prefer red were decreased by 2, what would be the percent of decrease?

2.) Tally the following scores in a frequency distribution. Do not use grouping.

3.) Make a frequency distribution of the following scores obtained by a basketball team.

b.) Calculate the percentage of frequency of each interval.

c.) Find the cumulative frequency for each interval.

d.) Calculate the percentage of each cumulative frequency relative to the total of the frequencies.

Dot Diagrams

When the data of a frequency distribution have not been grouped in intervals, they can be represented on a dot diagram. A dot diagram illustrates the pattern of a distribution. It clearly shows whether the data are spread out evenly or if they tend to cluster about any point.

To construct a dot diagram list the measurements, from lowest to highest, horizontally across the bottom of the graph. On the left side vertically list the frequencies or number of times that the measurements occur. For each time a measurement occurs place a dot in the column above the measurement.

Example:

67	68	69	70	70	71	71	71
72	72	72	74	74	74	74	76
76	76	76	80	80	80	84	85

Solution:

Exercises:

1.) Twenty workers were rated on a scale of 1 to 10 for efficiency. Construct a dot diagram to represent the following ratings: 7, 8, 9, 4, 5, 5, 7, 10, 6, 8, 7, 7. 5, 6, 9, 6.

2.) Draw a dot diagram to represent the following scores received on a spelling test: 98, 100, 78, 75, 68, 62, 75, 80, 82, 94, 80, 72, 75, 85, 85, 80, 70, 82, 78, 78, 72, 70, 90, 65.

3.) The distribution of heights of fifteen children is given below. Show the distribution on a dot diagram.

Histograms

Example:

72	82	56	73	87	89	72	86	88	76
86	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

Solution:

(figure available in print form)

Exercises:

1.) Construct a histogram for the following scores earned by a group of high school students on a Scholastic Aptitude Examination.

2.) The weights of 40 football players are as follows:

b.) Make a histogram for the given data.

Frequency Polygon

Example:

Scores	Midpoint	Frequency
96-100	98	3
9195	93	3
8690	88	4
8185	83	6
7680	78	8
7175	73	5
6670	68	3
6165	63	2
5660	58	2

Solution:

Exercises:

1.) The following table shows the weekly wages earned by workers in a local hospital:

b.) Construct a frequency polygon for the given data.

2.) A baseball team made the following number of hits in a recent game:

b.) Make a histogram for the given data.

c.) Construct a frequency polygon for the given data.

3.) The students in an English class received the following scores on a test:

b.) Draw a histogram to represent the scores.

c.) Construct a frequency polygon for the given data.

Cumulative Frequency Polygon

Exam ple:

86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

Solution:

		Cumulative	% of Cumulative
Scores	Frequency	Frequency	Frequency
96-100	3	40	100.0
9195	4	37	92.5
8690	8	33	82.5
8185	7	25	62.5
7680	6	18	45.0
7175	5	12	30.0
6670	3	7	17.5
6165	2	4	10.0
5660	2	2	5.0

(figure available in print form)

For some purposes the cumulative frequency polygon is very valuable. On the right side of the polygon is a scale of percent that parallels the scale of cumulative frequency. On the percent scale you read 25 corresponding to an abscissa of 72. This means that 25% of the scores were 72 or lower. The figure 72 is called the 25th percentile. The nth percentile is that score below which n percent of the scores in the distribution will fall.

To find the score that corresponds to a percentile on the graph, draw a horizontal line through the desired percent to intersect the cumulative frequency polygon. From the point of intersection draw a vertical line to the x-axis. The score at the point of intersection of the vertical line and the x-axis corresponds to the required percentile.

The fiftieth percentile is the median or middle score in a set of measurements. The 25th percentile is called the lower quartile, and the 75th percentile is the upper quartile.

Exercises:

1.) During one week a dealer sold the following number of cars: Monday 12, Tuesday 15, Wednesday 5, Thursday 6, Friday 10, Saturday 12.

b.) Make a frequency polygon to represent the given data.

c.) Draw a cumulative frequency polygon to represent the given data.

2.) The heights in inches of 50 high school students are:

b.) Construct a histogram to represent the data.

c.) Construct a cumulative frequency polygon.

d.) Find the median height. Find the upper and lower quartiles.

e.) Determine the 8Oth percentile.

3.) Forty students have the following IQ scores:

b.) Construct a cumulative frequency polygon.

c.) Determine the median IQ score and the 7Oth percentile.

Measures of Central Tendency

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, and 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.

Example 1:

Solution:

Example 2:

Solution:

Example 3:

Solution:

Another measure of central tendency is the median. When the elements of a set of numbers have been arranged in ascending order, the number in the middle of the set is the median of the set. The median divides the set of data into two equal parts. On a cumulative frequency polygon the median is the 50th percentile. To determine which element of a set is the middle number, use the following formula:

Middle Number = (Total Number of Elements + 1); = 2

If the set contains an even number of elements, the median is the average of the two middle numbers.

Example 1:

Solution:

Example 2:

Solution:

(10 + 1) Ö 2 = 11 Ö 2 = 5.5

The two middle numbers of the set are the fifth and sixth numbers: 80 and 82.

(80 + 82) Ö 2 = 162 Ö 2 = 81

The median score is 81.

A third, and most widely used, measure of central tendency is the arithmetic mean. The arithmetic mean is the average of a set of numbers. It is usually denoted by the symbol x. To calculate the arithmetic mean of a set of numbers, add the members of the set and divide the sum by the number of items in the set.

Example:

Solution:

The arithmetic mean of the set is 20.

Sometimes an item appears more than once in a set of measures. 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.

Example:

Solution:

Total Number of Items = 3 + 4 + 2 + 2 + 1 = 12

Sum of Products $dv$ Total # of Items = 300 $dv$ 12 = 25

The arithmetic mean is 25.

When the data have been arranged in intervals in a frequency distribution, the arithmetic mean is obtained in the following manner:

1.) Multiply the midpoint of each interval by the frequency of the interval.

2.) Find the sum of the products obtained in step 1.

3.) Divide the sum obtained in step 2 by the total number of items in the distribution.

n

x = 1/ni å1 xifi

x ; arithmetic mean	xi = midpoint of the interval
n = number of items in	fi = frequency of the interval
the distribution	= sum

Example:

Scores	Midpoint	Frequency	xifi
96-100	98	3	294
91-95	93	4	372
86-90	88	8	684
81-85	83	7	581
76-80	78	6	468
71-75	73	5	365
66-70	68	3	204
61-65	63	2	126
56-60	58	2	116

Solution

 40

åxifi= 294 + 372 + 684 + 581 + 468 + 365+ i= 1  204 + 126 + 116 = 3210

40

x = 1/n å xifi 1/40 x 3210 = 3210/40 = 80.25

i = 1

The arithmetic mean of the distribution is 80.25.

Exercises:

1.) Ten employees of a department store earn the following weekly wages: $200, $150, $160, $125, $160, $150, $180, $130, $170 $150

b.) What is the median wage?

c.) Find the mode.

2.) Write mean, median, or mode to complete the sentence.

3.) Find the mean, the median and the mode for each set of numbers.

b.) 13, 19, 12, 18, 24, 10.

c.) 125, 132, 120, 118, 128, 126, 120.

d.) 8, 4, 6, 4, 10, 4, 10.

4.) Find the arithmetic mean of the following numbers:

5.) The salaries of thirty people are listed below.

b.) What is the mode of the salaries?

c.) What is the median salary?

d.) What is the mean salary?

Measures of Disperseion

Range = Highest Number—Lowest number

Example:

Solution:

15Ð1 = 14. The range of the set is 14.

Another way of indicating the dispersion of scores is in terms of their deviations from the mean. This method is known as standard deviation and tells how scores tend to scatter about the mean of a set of data. If the standard deviation is small the scores tend to cluster closely about the mean. If the standard deviation is large, there is a wide scattering of scores about the mean. Standard deviation is represented by the symbol s and may be computed by the formula:

Standard Deviation = s=

(figure available in print form)

where x is a score, x is the mean, n is the number of scores, and means “the sum of”.

Six steps are used to find standard deviation:

1.) List each score (x) in the set of data.

2.) Compute the mean (x) for the data.

3.) Subtract the mean from each score (xÐx). The result is the deviation of each score from the mean.

4.) Square the deviations.

5.) Find the average of the squares of the deviations by dividing the sum of the squares of the deviations by the number of scores in the distribution.

6.) Take the square root of this average. The result is the standard deviation.

Example:

Solution

The standard deviation is a number that is used to compare scores in a distribution. If the mean of a group of test scores is 75, and the standard deviation is 10, a person who receives a score of 85 is one standard deviation above the mean. If the mean of another group of test scores is 80, and the standard deviation is 3, a person who receives a score of 83 is one standard deviation above the mean. This person has done equally well, with respect to the other class members, as the person who received 85 on the first test.

Exercises:

1.) Compute the range for the following sets of scores:

b.) 113, 98, 107, 102, 123, 110

c.) 72.9, 75.6, 74.3, 86.1, 80, 82.7

d.) 56, 72, 98, 64, 87, 91, 22

2.) Compute the standard deviations for the following sets of scores:

b.) 111, 98, 107, 103, 126

c.) 72.9, 75.6, 74.3, 86.1, 80, 82.7

3.) On an arithmetic test the mean was 78 and the standard deviation was 8. How many standard deviations from the mean was each of the following scores? 86, 74, 94, 80, 98, 70, 62

Bibliography for Teachers

Reading List for Students