Answers and Rounding Off

A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers will be rounded off in this manner.

It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it is more helpful to leave an answer as an unreduced fraction.

Levels of Measurement

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):

• Nominal scale level
• Ordinal scale level
• Interval scale level
• Ratio scale level

Data that is measured using a nominal scale is qualitative. Categories, colors, names, labels and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful.

Smartphone companies are another example of nominal scale data. Some examples are Sony, Motorola, Nokia, Samsung and Apple. This is just a list and there is no agreed upon order. Some people may favor Apple but that is a matter of opinion. Nominal scale data cannot be used in calculations.

Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data.

Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are “excellent,” “good,” “satisfactory,” and “unsatisfactory.” These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations.

Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point.

Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like -10° F and -15° C exist and are colder than 0.

Interval level data can be used in calculations, but one type of comparison cannot be done. 80° C is not four times as hot as 20° C (nor is 80° F four times as hot as 20° F). There is no meaning to the ratio of 80 to 20 (or four to one).

Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded.

The data can be put in order from lowest to highest: 20, 68, 80, 92.

The differences between the data have meaning. The score 92 is more than the score 68 by 24 points.
Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 56332475235654435253.

[link] lists the different data values in ascending order and their frequencies.

Frequency Table of Student Work Hours

DATA VALUE	FREQUENCY
2	3
3	5
4	3
5	6
6	2
7	1

A frequency is the number of times a value of the data occurs. According to [link], there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Frequency Table of Student Work Hours with Relative Frequencies

DATA VALUE	FREQUENCY	RELATIVE FREQUENCY
2	3	\(\frac{3}{20}\) or 0.15
3	5	\(\frac{5}{20}\) or 0.25
4	3	\(\frac{3}{20}\) or 0.15
5	6	\(\frac{6}{20}\) or 0.30
6	2	\(\frac{2}{20}\) or 0.10
7	1	\(\frac{1}{20}\) or 0.05

The sum of the values in the relative frequency column of [link] is \(\frac{20}{20}\)
, or 1.

Cumulative relative frequency is the accumulation of the previous relative
frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to
the relative frequency for the current row, as shown in [link].

Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies

DATA VALUE	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
2	3	\(\frac{3}{20}\) or 0.15	0.15
3	5	\(\frac{5}{20}\) or 0.25	0.15 + 0.25 = 0.40
4	3	\(\frac{3}{20}\) or 0.15	0.40 + 0.15 = 0.55
5	6	\(\frac{6}{20}\) or 0.30	0.55 + 0.30 = 0.85
6	2	\(\frac{2}{20}\) or 0.10	0.85 + 0.10 = 0.95
7	1	\(\frac{1}{20}\) or 0.05	0.95 + 0.05 = 1.00

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

[link] represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

Frequency Table of Soccer Player Height

HEIGHTS (INCHES)	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
59.95–61.95	5	\(\frac{5}{100}\) = 0.05	0.05
61.95–63.95	3	\(\frac{3}{100}\) = 0.03	0.05 + 0.03 = 0.08
63.95–65.95	15	\(\frac{15}{100}\) = 0.15	0.08 + 0.15 = 0.23
65.95–67.95	40	\(\frac{40}{100}\) = 0.40	0.23 + 0.40 = 0.63
67.95–69.95	17	\(\frac{17}{100}\) = 0.17	0.63 + 0.17 = 0.80
69.95–71.95	12	\(\frac{12}{100}\) = 0.12	0.80 + 0.12 = 0.92
71.95–73.95	7	\(\frac{7}{100}\) = 0.07	0.92 + 0.07 = 0.99
73.95–75.95	1	\(\frac{1}{100}\) = 0.01	0.99 + 0.01 = 1.00
	Total = 100	Total = 1.00

The data in this table have been grouped into the following intervals:

• 59.95 to 61.95 inches
• 61.95 to 63.95 inches
• 63.95 to 65.95 inches
• 65.95 to 67.95 inches
• 67.95 to 69.95 inches
• 69.95 to 71.95 inches
• 71.95 to 73.95 inches
• 73.95 to 75.95 inches

In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.