Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.
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 (categorical). 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 name does not make any sense.
Healthcare companies are another example of nominal scale data. The data are the names of the companies that provide health services, but there is no agreed upon order of these companies, even though people may have personal preferences. 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 exercise habits for adults in the United States. The top five exercise habits 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 exercise satisfaction survey where the responses to questions about the exercise class 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 emotional regulation survey scores are 80, 68, 20 and 92 (out of a possible 100 points).
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 ate per day. Their responses, in number of times, are as follows: 5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.
Table \(\PageIndex{1}\) lists the different data values in ascending order and their frequencies.
| 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 Table \(\PageIndex{1}\):, there are three students who ate 2 meals, five students who ate three meals, 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.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY | |
|---|---|---|---|
| 2 | 3 | \(\dfrac{3}{20}\) or 0.15 | |
| 3 | 5 | \(\dfrac{5}{20}\) or 0.25 | |
| 4 | 3 | \(\dfrac{3}{20}\) or 0.15 | |
| 5 | 6 | \(\dfrac{6}{20}\) or 0.30 | |
| 6 | 2 | \(\dfrac{2}{20}\) or 0.10 | |
| 7 | 1 | \(\dfrac{1}{20}\) or 0.0 | |
The sum of the values in the relative frequency column of Table \(\PageIndex{2}\) is \(\dfrac{20}{20}\), 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 Table \(\PageIndex{3}\).
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 2 | 3 | \(\dfrac{3}{20}\) or 0.15 | 0.15 |
| 3 | 5 | \(\dfrac{5}{20}\) or 0.25 | 0.15+0.25=0.40 |
| 4 | 3 | \(\dfrac{3}{20}\) or 0.15 | 0.40+0.15=0.55 |
| 5 | 6 | \(\dfrac{6}{20}\) or 0.30 | 0.55+0.30=0.85 |
| 6 | 2 | \(\dfrac{2}{20}\) or 0.10 | 0.85+0.10=0.95 |
| 7 | 1 | \(\dfrac{1}{20}\) or 0.0 | 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.
NOTE
Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.
Table \(\PageIndex{4}\) represents the heights, in inches, of a sample of 100 semiprofessional soccer players.
| HEIGHTS (INCHES) |
FREQUENCY | RELATIVE FREQUENCY |
CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 59.95-61.95 | 5 | \(\dfrac{5}{100}\)=0.05 | 0.05 |
| 61.95-63.95 | 3 | \(\dfrac{3}{100}\)=0.03 | 0.05+0.03=0.08 |
| 63.95-65.95 | 15 | \(\dfrac{15}{100}\)=0.15 | 0.08+0.15=0.23 |
| 65.95-67.95 | 40 | \(\dfrac{40}{100}\)=0.40 | 0.23+0.40=0.63 |
| 67.95-69.95 | 17 | \(\dfrac{17}{100}\)=0.17 | 0.63+0.17=0.80 |
| 69.95-71.95 | 12 | \(\dfrac{12}{100}\)=0.12 | 0.80+0.12=0.92 |
| 71.95-73.95 | 7 | \(\dfrac{7}{100}\)=0.07 | 0.92+0.07=0.99 |
| 73.95-75.95 | 1 | \(\dfrac{1}{100}\)=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:
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.
From Table \(\PageIndex{4}\), find the percentage of heights that are less than 65.95 inches.
- Answer
-
If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are 5 + 3 + 15 = 23 players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then \(\dfrac{23}{100}\) or 23%. This percentage is the cumulative relative frequency entry in the third row.
From Table \(\PageIndex{4}\), find the percentage of heights that fall between 61.95 and 65.95 inches.
- Answer
-
Add the relative frequencies in the second and third rows: 0.03 + 0.15 = 0.18 or 18%.
Use the heights of the 100 semiprofessional soccer players in Table \(\PageIndex{4}\). Fill in the blanks and check your answers.
- The percentage of heights that are from 67.95 to 71.95 inches is: ____.
- The percentage of heights that are from 67.95 to 73.95 inches is: ____.
- The percentage of heights that are more than 65.95 inches is: ____.
- The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.
- What kind of data are the heights?
- Describe how you could gather this data (the heights) so that the data are characteristic of all semiprofessional soccer players.
Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
- Answer
-
29% 36% 77% 87 quantitative continuous get rosters from each team and choose a simple random sample from each
Table \(\PageIndex{7}\) contains data for the age, in years, for 70 healthcare patients.
| Age (Years) | Number of Healthcare Patients |
|---|---|
| 24 | 2 |
| 25 | 1 |
| 26 | 3 |
| 27 | 0 |
| 28 | 4 |
| 29 | 6 |
| 30 | 11 |
| 31 | 12 |
| 32 | 7 |
| 33 | 8 |
| 34 | 6 |
| 35 | 10 |
Problem
Answer the following questions.
- What is the cumulative frequency for age in years between 30 and 35 (inclusive)?
- What is the relative frequency for 30 years old?
- What is the relative frequency for 30 years old or less?
- What is the relative frequency for 25 years old or more?
- Answer
-
Cumulative frequency is 54. 11/70 or 0.157 or 15.7% 27/70 or 0.386 or 38.6% 68/70 or 0.971 or 97.1%
Solution
- Cumulative frequency is 54.
- 11/70 or 0.157 or 15.7%
- 27/70 or 0.386 or 38.6%
- 68/70 or 0.971 or 97.1%