1.4 Levels of Measurement, Frequency, and Relative Frequency

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

Nominal Scale

Data that is measured using a nominal scale is qualitative and has no natural order. Categories such as colors, names, labels, and favorite foods along with yes or no responses are examples of nominal scale 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.

Ordinal Scale

Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data categories 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.

Interval Scale

Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a measurable and meaningful 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 would could find meaning in the difference between two values. For example, 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 between data values cannot be done. For example, 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 this type of comparison. A ratio of 80 to 20 (or four to one) does not make sense.

Ratio Scale

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 meaningful 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 and 0 is the absolute lowest value. We can compare scores in a meaningful way as well, 80 is four times 20. The score of 80 is four times better than the score of 20.

Rounding and Accuracy

When we work with data, we will calculate summary values that may involve multiplying, dividing, or finding square roots. These sorts of calculations can result in lots of digits showing up after a decimal point. Because of this, some calculations generate numbers that are artificially precise. It is not necessary to report a calculated number to eight decimal places when the data values that generated the number were only accurate to the nearest tenth. Round off your final calculated answers to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth. This will help avoid thinking numbers are more accurate than they are and rounding error exists only in a lower place value position.

Frequency

Consider the following example.  Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3.

The following table lists the different data values in ascending order and their frequencies.

A frequency is the number of times a value of the data occurs. According to the table, 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. So a relative frequency of can be expressed equivalently as 0.2, which is equivalent to 20%.

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 the table below.

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

DATA VALUE	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
2	3		0.15
3	5		0.15 + 0.25 = 0.40
4	3		0.40 + 0.15 = 0.55
5	6		0.55 + 0.30 = 0.85
6	2		0.85 + 0.10 = 0.95
7	1		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.