Math 365, Elementary Statistics

Lesson 1: The Language and Terminology

Introductionback to top

Most people think of statistics as the study of the numerical features of a subject/population. It means the same to statisticians, but also emphasizes the methods of collecting data, summarizing and presenting data, and drawing inferences from data.

We all see on TV how political pundits justify opposing points of view by presenting statistics from respectable sources. How could something be a science when it justifies two opposing points of view? The answer is that statistics has a scientific basis but it can be misrepresented in use.

Example. During the saga of President Clinton's impeachment, we observed the following:

  1. One pundit says that, according to statistics, the majority of Americans think that character matters.
  2. The other pundit says, also according to statistics, that the majority of Americans think the president is doing a good job.

The implication here is that one of them was "wrong." But the science of statistics says that both were correct. Data was collected and analyzed, and it was found that the majority of Americans think that character matters and that the majority of Americans think the president is doing a good job. It does not matter to the science of statistics which one of the statistically established facts you or I want to believe.

Another point about the nature of statistics as a science is that it is not a deterministic science. It does not have laws like force is equal to mass times acceleration. Statements in statistics come with a probability (i.e., quantified chance) of being correct. When a weatherman says that it will rain today he means that there is, say, a ninety five percent chance that it will rain today. Roughly, this means that if he makes the same prediction one hundred times he will be correct 95 times, and it will not rain the other 5 days. The problem is that sometimes a weatherman will hide the information that there is a 95 percent chance only. Such information hiding is sometimes done for simplicity.

Before I conclude this introduction, let me tell you an interesting anecdote about the development of this subject. When the proposal to establish the Indian Statistical Institute in Calcutta was considered by the government of India in the early part of the last century, some critics said, then why not an institute in astrology? At the inception of statistics as a science there was a lot of skepticism about its scientific validity. Those days are gone, and statistics is not likened to astrology any more! Statistics is a well-founded and precise science. It is a nondeterministic science in nature; it makes precise probabilistic statements only.

In this course we will be talking about two branches of statistics. The first one is called descriptive statistics and deals with methods of processing, summarizing, and presenting data. The other part deals with the scientific methods of drawing inferences and forecasting from the data, and is called inferential or inductive statistics.

In the rest of this lesson and the next we deal with descriptive statistics, which includes the presentation of data in the form of tables, graphs, and computations of various averages of data.

1.1 Basic Definitions and Conceptsback to top

In statistics we use a small representative "sample" to study a big "population." The reason for this is the cost or even the impossibility of studying the whole population.

Population and Sample

Definitions. A complete collection of data on the group under study is called the population or the universe.

A member of the population is called a sampling unit. Therefore, the population consists of all its sampling units.

A Sample is a collection of sampling units selected from the population.

Most often, we will work with numerical characteristics (like height, weight, and salary) of a group. So usually the population is a large collection of numbers and the sample is a small subset of the population.

Example. Suppose we are studying the daily rainfall in Lawrence. Since daily rainfall could be from 0 inches to anything above 0, the population here is all nonnegative numbers (i.e., the interval [0, ∞)). A sample from this population would be the observed amount of daily rainfall in Lawrence on some number of days. A sample of size 11 would be the observed daily rainfall in Lawrence on 11 days.

Variables

Many definitions of variables are available in standard textbooks. For our purpose the following definition will suffice.

Definition. A variable is a rule or a formula or a mechanism that associates a value with each member of the population. So, given a member w, a variable X assigns a value X(w) to w. For us X(w) will be a characteristic (like height, weight, time, salary) of the population.

Example. Suppose we are studying the KU student population. The population is the whole collection of KU students. A KU student is a sample unit. If GPA is the "characteristic" that we are studying, then X = the GPA of a student is a variable. So, given a student, X has a value. For example:

X(Donald Smith) = 3.25,     X(Sam Donaldson) = 3.11,
X(Karen Currie) = 3.89,     X(King Who) = 2.13

On the other hand, if GENDER is the "characteristic" that we are studying, then Y = gender of a student is a variable. So, given a student, Y has a value. For example:

Y(Donald Smith) = Male,     Y(Sam Donaldson) = Male,
Y(Karen Currie) = Female ,     Y(King Who) = Male

If HEIGHT is the characteristic that we are studying, then Z = height of students is a variable.

To give another example, if credit hours completed is the characteristic studied, T = the number of course credit hours completed so far by a student is a variable.

Similarly, given any other characteristic like weight, annual income, annual expenditure, you can construct a variable for this population.

A variable that takes numerical values is called a quantitative variable. So, the variables X, Z, and T above are quantitative variables, while Y is not. A variable that takes non-numerical values is called a qualitative variable. So, the variable Y above is a qualitative variable. We will mostly be concerned with quantitative variables.

We discuss two types of quantitative variables: continuous and discrete variables. A quantitative variable that can assume any numerical value over an interval is called a continuous variable. Since Z above can (hypothetically) assume any value between 0 to 100 inches, Z is a continuous variable. T assumes only integer values and is therefore not a continuous variable.

A different way to understand a discrete variable is that the possible values of the variable can be written down (or can be counted) in a (finite or infinite) list. We say that the values of a discrete variable are countable.

A quantitative variable is called a discrete variable if its possible values consist of breaks between successive values. If a variable assumes only a finite number of values, then it is also called a finite variable. Otherwise the variable is called an infinite variable. A finite variable is definitely a discrete variable. The variable T above is a discrete variable.

Examples of Continuous and Discrete Variables

  1. The examples of continuous variables are weight, length, volume, area, and time.
  2. For this course, examples of discrete variables are always the number of something—number of typos, number of road accidents, number of phone calls.

Parameters and Statistics

Definition 1. Given a set of data, any numerical value computed from the data using a formula or a rule is called a quantitative measure of the data.

Definition 2. A quantitative measure of a population data is called a parameter. In other words, parameters belong to the whole population and are computed (if feasible) from the WHOLE population data. Examples: the average GPA of all KU students, the height of the tallest student in KU, the average income of the entire KU student population.

One way to study a population is to know some of the parameters of the population. Unfortunately, computing such parameters could be expensive or even impossible. Essentially, parameters are unknown and the main game of statistics is to try to estimate parameters on the basis of small samples collected from the population.

Definition 3. A quantitative measure of a sample data is called a statistic. So, any constant that we compute from a sample is a statistic. We use these statistics to estimate the parameters of the population. For example, the average height computed from a sample is a reasonable estimate for the (parameter) average height of the KU student population. Obviously, we do not expect the value of the statistic to be exactly equal to the parameter value. Hopefully, the error will be small or will exceed our tolerable limit very rarely (say once in a 100 trials).

Why do we need a statistic?

Sometimes it will be impossible to know the actual value of a parameter. For example, let μ be the mean length of the life of light bulbs produced by a company. In this case, the company cannot test all the bulbs it produces to find a mean length. So, the best it can do is to test a few bulbs, compute the sample mean length (a statistic) of the life of these bulbs and use it as an estimate for the mean length (parameter μ) of the life for all the bulbs it produces.

Definition 4. The data that has not been processed or organized in any form is called raw data. When the data is arranged in an increasing or decreasing order, then it is called an array. The range of the data is the difference between the largest and the smallest value of the data.

range = highest value - lowest value.

1.2 Frequency Distributionback to top

In this section we talk about representation of data organized in tabular form. Such a representation is called a frequency distribution. We are mostly concerned with numerical data (i.e., quantititative data), but also consider some non-numerical data (i.e., qualitative data).

Example. (from Khazanie, p. 18) The following is data on the blood group of 36 patients in a hospital:

O A B O A A A O O
O A O A B O O O AB
B A A O O A A O AB
O A A B A O A O O

We have four types of blood groups, namely, O, A, B, AB. Each of these blood groups may be referred to as a "class." The frequency of a class is defined as the number of data members that belong to that class. For example, the frequency of the class O is 16; the frequency of class A is 14. A table that lists the classes and the corresponding frequency is called the frequency distribution of this qualitative data. Following is the frequency distribution of this data:

Blood Group Frequency
O 16
A 14
B 4
AB 2
Total 36


Ungrouped Data

For the quantitative data, we consider two types of frequency table. When we are working with a large set of data we group that data into a few classes and construct a "frequency table," which we will discuss later. If the data set is small or if the number of values that appear in the data is small we need not group the data. Instead, we make a list of all the data members and give the corresponding frequency for each data member in a table. The number of times a data member (i.e., value) appears in the data is called the frequency of the data member. A list that presents the data members and the corresponding frequency in a tabular form is called a frequency table or frequency distribution. The relative frequency and percentage frequency of a data member x are defined as follows:


relative frequency of x = frequency of x
total # of data points
and
percentage  frequency of x = frequency of x
total # of data points
100.

The frequency table may also contain the relative and percentage frequency. Since we did not group the data into a few classes, we call this the frequency distribution of the ungrouped data.

Example 1.2.1 To estimate the mean time taken to complete a three-mile drive by a race car, the race car did several time trials, and the following sample of times taken (in seconds) to complete the laps was collected:

50 48 49 46 54 53 52 51 47 56 52 51
51 53 50 49 48 54 53 51 52 54 54 53
55 48 51 50 52 49 51 53 55 54 50  

Note that there are 35 observations here. So we say that the size of the sample (or data) is 35. Also the values present are 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56. Since there are only 11 distinct values present we can make a frequency table for the ungrouped data. The following is the frequency distribution of this ungrouped data:

Time
(in seconds)
Frequency Relative
Frequency
Percentage
Frequency
46 1 1/35 2.86
47 1 1/35 2.86
48 3 3/35 8.57
49 3 3/35 8.57
50 4 4/35 11.43
51 6 6/35 17.14
52 4 4/35 11.43
53 5 5/35 14.29
54 5 5/35 14.29
55 2 2/35 5.71
56 1 1/35 2.86
Total 35 1 100

Grouped Data

When we are working with a large set of data that has too many distinct class member (i.e., values) then we group the whole set of data into a few class intervals and give the corresponding "frequency" of the class. When the data is presented in this way, the data is called grouped data. The number of data members that fall in a class interval is called the class frequency and the relative and percentage frequencies are computed by the same formula as above. A list that gives various class intervals and the corresponding class frequencies in a tabular form is called a class frequency table or class frequency distribution of the data. The frequency distribution may also include the relative and percentage frequencies.

Grouped Data and Loss of Information

Sometimes it is convenient or necessary to group data into class intervals and construct a class frequency distribution. This is the case when there are too many distinct numbers present in the data—too many even to fit into a simple table on a page for presentation. In such situations, we group the data in a few class intervals. While class frequency distribution is very good for presentation and convenient for other reasons, we lose a lot of information in this process. There is no way we can recover the original data from the class frequency distribution.

Given a set of data, a good question would be, How many class intervals should we have? The answer is that it should not be too few nor should it be too many. If we take too few (say one), then all the information will be lost. On the other hand, if we take too many, we will have the problem of having to work with ungrouped data. (In this course we will always tell you how many classes to take.) Although sometimes it may be necessary to take class intervals of varying width, in this course we only consider classes of equal class width.

Steps to Construct Frequency Distribution

  1. Range: Pick a suitable number L less than or equal to the smallest value present in the data. Pick a suitable number H greater than or equal to the highest value present in the data. The range R that we consider is R = H - L.
  2. Number of Classes: Decide on a suitable number of classes. (In this course we will tell you the number of classes.)
  3. Class Width: We have
    class width = w = R
    Number of classes

    We will pick L, H, and the number of classes so that class width is a "round number."
  4. Classes: We divide our interval [L:H] into subintervals, to be called classes, as

    [L,L+w],[L+w,L+2w],[L+2w, L+3w], ...,[H-w,H]

    Since this definition creates an ambiguous situation in which a data value may fall into two classes, we need a convention to address this situation.

  5. Frequency: Find the frequency for each of the classes. You can use an advanced calculator or some software (like Excel) to count frequencies.

A few more important definitions. The above intervals are called class intervals. The w above is called the class size or width. The lower end of the class is called lower limit and the upper end of the class is called upper limit. The class mark is the midpoint of the class, defined as follows:


class mark = lower limit of class+ upper limit of class
2
.

A class limit is also called a class boundary. I took a slightly different approach when I defined the classes, so that for us class limits and class boundaries are the same. Although all the approaches are essentially the same, many slightly different approaches are possible depending on the situation.

Example 1.2.2 The following is the weight (in ounces), at birth, of a certain number of babies.

74 105 124 110 119 137 96 110 120 115 140
65 135 123 129 72 121 117 96 107 80 91
74 123 124 124 134 78 138 106 130 97 145
93 133 128 96 126 124 125 127 62 127 92
95 118 126 94 127 121 117 124 93 135 156
143 125 120 147 138 72 119 89 81 113 91
133 127 138 122 110 113 100 115 110 135 141
97 127 120 110 107 111 126 132 120 108 148
143 103 92 124 150 86 121 98 74 85 99

We will construct a class frequency table of this data by dividing the whole range of data into class intervals.

Solution: Note that the lowest value is 62 and the highest value is 156. We take L = 60, H = 160, so R = H-W = 100. We made such a choice of L and H, precisely so that R = 100 is a "nice" number. Now we decide to have 5 class intervals and so w = R/5 = 20. According to what I said above, our classes should be : [60, 80], [80,100], [100,120], [120,140], [140, 160]. But if we do so then there is a risk that some data members (like 80, 100, 120, 140) will fall in two classes. One way to avoid this is to add .5 to all the class boundaries. So, our classes are [60.5, 80.5], [80.5, 100.5], [100.5, 120.5], [120.5, 140.5], [140.5, 160.5].

So the frequency distribution is as follows:

Classes Frequency Relative
Frequency
Percentage
Frequency
60.5 - 80.5 9 9/99 9.09
80.5 - 100.5 20 20/99 20.20
100.5 - 120.5 25 25/99 25.26
120.5 - 140.5 37 37/99 37.38
140.5 - 160.5 8 8/99 8.08
Total 99 1 100

1.3 Pictorial Representation of Databack to top

Another way to represent data is to use pictures and graphs. We see such pictorial representation in newspapers and other sources every day. Pictorial representation is particularly important when you have to represent data to people with limited technical background, like newspaper readers or a governmental or congressional body.


The Pie Chart

The pie chart is a commonly used pictorial representation of data. When you do your tax return every year, you find a few pie charts in the instruction book for form 1040. These charts show what proportion/percentage of each tax dollar goes for particular expenses. I reproduced the following pie charts from the 1040 instruction book of 1999.

a pie chart showing how tax dollar outlays are distributed a pie chart showing how tax dollar income is distributed
Pie charts are self explanatory; we do not need to discuss them further.

The Histogram

Among pictorial representations, the most useful in this course is the histogram. The histogram of data is the graphical representation of the frequency distribution of the data, where we plot the variable on the horizontal axis and above each class interval, we erect a bar of the height equal to the frequency of the class. Such a histogram is called a frequency histogram.

If, instead, we erect bars of height equal to the relative frequency, then the graph is called a relative frequency histogram. Similarly, we can construct a percentage frequency histogram.

The following is a histogram.

a histogram

We have decided to avoid unequal class lengths, which makes our discussion of the histogram fairly simple.

Remark. Take a look at the Stem and Leaf Diagram discussed in any textbook.

Example 1.3.1. Following is the frequency table of data on height (in inches) of some babies at birth. Sketch the histogram of the following data:

Height Frequency
16-17 3
17-18 8
18-19 34
19-20 60
20-21 72
21-22 18

The Cumulative Frequency Distributions

For a given value x of a variable, the cumulative frequency of the data, for x, is the number of data members that are less than or equal to x.

Definition. Given a frequency distribution of some data, for a class boundary x, the cumulative frequency is the sum of all the class frequenies less or equal to x. The cumulative frequency distribution is a table that gives the cumulative frequencies against some x values (for us the class boundaries). We also define cumulative relative frequency and cumulative percentage frequency as follows:



cumulative relative frequency of x =
cumulative frequency of x
total # of data points

cumulative percentage frequency of x= cumulative frequency
total # of data points
100

Example 1.3.2 Once again we consider the data on birth weight of babies in Example 1.2 that we discussed in the last section. A cumulative frequency distribution can be constructed from the frequency distribution.

Solution: We have seen the frequency distribution before. The following is the cumulative distributions:

Weight Cumulative
Frequency
Relative-Cumulative
Frequency
Cumulative
Percentage
Frequency
60.5 0 0 0
80.5 9 9/99 9.09
100.5 29 29/100 29.29
120.5 54 54/99 54.55
140.5 91 91/99 91.92
160.5 99 1 100


The Ogive

Definition. The ogive is a line graph, where we plot the variable on the horizontal axis and the cumulative frequency on the vertical axis. If we plot the cumulative relative frequency on the vertical axis, then the line graph is called the relative frequency ogive.

relative frequency ogive graph


Use of Calculators

Because we will be using calculators (TI-83) extensively in this course, let me explain how you enter data in the TI-83.

Use of Calculators (TI-83):
Enter Your Data:
  1. Press the button "stat."
  2. Select "Edit" in the Edit menu and enter.
  3. You will find 6 lists named L1, L2, L3, L4, L5, L6.
  4. Let's say you want to enter your data in L1. If L1 has some data, you clear it by pressing the stat button and selecting ClrList in the Edit menu. ClrList appears then type L1 and hit enter. To type "L1" on your TI-83 simply press 2nd then 1.
  5. Once L1 is cleared, you select Edit in the Edit menu and enter.
  6. Now type in your data; enter one by one.

It is not easy to construct a frequency table of a data set unless you are systematic. Traditionally, we used "tally marks" to count the frequency. Now you can use some software programs (e.g., Excel). Let me show you a method, using a calculator (TI-83).

  1. Press "stat."
  2. To input data, enter "edit."
  3. Enter your data (say in L1).
  4. Press "stat."
  5. Enter "sortA" L1.
  6. Press "stat" and then enter "edit." On L1 you will see that the data is sorted in an increasing order.
  7. Now you can count the frequencies.

Problems on 1.2: Frequency Distributionback to top


Exercise 1.2.1
To estimate the mean time taken to complete a three-mile drive by a race car, the race car did several time trials, and the following sample of times taken (in seconds) to complete the laps was collected:

50 48 49 46 54 53 52 51 47 56 52 51
51 53 50 49 48 54 53 51 52 54 54 53
55 48 51 50 52 49 51 53 55 54 50  

The following is the frequency distribution of this ungrouped data:

Time
(in seconds)
Frequency Relative
Frequency
Percentage
Frequency
46 1 1/35 2.86
47 1 1/35 2.86
48 3 3/35 8.57
49 3 3/35 8.57
50 4 4/35 11.43
51 6 6/35 17.14
52 4 4/35 11.43
53 5 5/35 14.29
54 5 5/35 14.29
55 2 2/35 5.71
56 1 1/35 2.86
Total 35 1 100

Construct a histogram.

Exercise 1.2.2. The following is the weight (in ounces), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.

94 105 124 110 119 137 96 110 120 115 119
104 135 123 129 72 121 117 96 107 80 80
96 123 124 124 134 78 138 106 130 97 134
111 133 128 96 126 124 125 127 62 127 96
116 118 126 94 127 121 117 124 93 135 112
120 125 120 147 138 72 119 89 81 113 100
109 127 138 122 110 113 100 115 110 135 120
97 127 120 110 107 111 126 132 120 108 148
133 103 92 124 150 86 121 98

Construct a class frequency table of this data by dividing the the whole range of data into class intervals:

[60.5-70.5], [70.5-80.5], [80.5-90.5], [90.5-100.5], [100.5-110.5], [110.5-120.5], [120.5-130.5], [130.5-140.5], [140.5-150.5]

Solution

Exercise 1.2.3. The following are the length (in inches), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.

18 18.5 19 18.5 19 21 18 19 20 20.5
19 19 21.5 19.5 20 17 20 20 19 20.5
18 18.5 20 19.5 20.75 20 21 18 20.5 20
21 19 20.5 19 20 19.5 17.75 20 19.5 20
20.5 17 21 18.5 20 20 20 18.5 19.5 19
18 20.5 18 20 19 19 19.5 20 20.75 21
17.75 19 18 19 20 18.5 20 19 21 19
19.5 20 20 19 19.5 20 19.5 18.5 20.5 19.5
20.25 20 19.5 19.5 20 20 20 21 20 19
18.5 20.5 21.5 18 19.5 18

Construct a frequency table for this data by dividing the whole range into class intervals:

[16-17], [17-18], [18-19], [19-20], [20-21], [21-22].

Note: If a data member falls on the boundary, count it in the right/upper class-interval.
Solution

Exercise 1.2.4. The following data represents the number of typos in a sample of 30 books published by some publisher.

156 159 162 160 156 162
159 160 156 156 160 162
156 159 162 156 162 158
160 158 159 162 158 158
162 160 159 162 162 160

Construct a frequency table (by sorting in your calculator). Also construct a histogram.
Solution

Exercise 1.2.5. Following is data on the hourly wages (paid only in whole dollars) in an industry.

9 11 8 9 10 11 7 10 12 13
7 11 8 11 14 9 10 9 11 7
13 13 14 12 9 8 12 14 15 9
9 7 12 7 12 7 7 11 13 9
11 9 9 9 10 14 11 12 14 7

Construct a frequency table (by sorting in your calculator). Also construct a histogram.
Solution

Exercise 1.2.6. Following is data on the hourly wages (paid only in whole dollars) of 99 employees in an industry.

7 11 7 11 10 9 10 10 12 13
7 8 11 11 14 9 7 9 11 7
9 13 12 14 7 8 7 14 15 9
9 7 11 9 12 9 12 11 14 9
12 13 7 9 10 14 11 12 13 7
15 15 16 16 15 16 11 7 18 19
15 16 15 15 16 16 17 16 16 13
15 15 16 15 16 15 15 17 16 12
16 15 15 16 15 15 19 8 16 17
16 16 15 16 16 16 13 12 8  

Construct a frequency table (by sorting in your calculator).