Describing Frequencies
What is a frequency?
The frequency is the number of times a particular value for a variable (data item) has been observed to occur.
How can we measure frequency?
The frequency of a value can be expressed in different ways, depending on the purpose required.
The absolute frequency describes the number of times a particular value for a variable (data item) has been observed to occur.
The simplest way to express a frequency is in absolute terms.
A relative frequency describes the number of times a particular value for a variable (data item) has been observed to occur in relation to the total number of values for that variable.
The relative frequency is calculated by dividing the absolute frequency by the total number of values for the variable.
How are relative frequencies expressed?
Ratios, rates, proportions and percentages are different ways of expressing relative frequencies.
A ratio compares the frequency of one value for a variable with another value for the variable.
The first value identified in a ratio must be to the left of the colon (:) and the second value must be to the right of the colon (1st value : 2nd value).
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the ratio of heads to tails is 12:8. Alternatively, the ratio of tails to heads is 8:12.
A rate is a measurement of one value for a variable in relation to another measured quantity.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the rate is 12 heads per 20 coin tosses. Alternatively, the rate is 8 tails per 20 coin tosses.
A proportion describes the share of one value for a variable in relation to a whole.
It is calculated by dividing the number of times a particular value for a variable has been observed, by the total number of values in the population.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the proportion of heads is 0.6 (12 divided by 20). Alternatively, the proportion of tails is 0.4 (8 divided by 20).
A percentage expresses a value for a variable in relation to a whole population as a fraction of one hundred.
The percentage total of an entire dataset should always add up to 100, as 100% represents the total, it is equal to the ‘whole’. A percentage is calculated by dividing the number of times a particular value for a variable has been observed, by the total number of observations in the population, then multiplying this number by 100.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the percentage of heads is 60% (12 divided by 20, multiplied by 100). Alternatively, the percentage of tails is 40% (8 divided by 20, multiplied by 100).
Describing Frequencies
What is a frequency?
The frequency is the number of times a particular value for a variable (data item) has been observed to occur.
How can we measure frequency?
The frequency of a value can be expressed in different ways, depending on the purpose required.
The absolute frequency describes the number of times a particular value for a variable (data item) has been observed to occur.
The simplest way to express a frequency is in absolute terms.
A relative frequency describes the number of times a particular value for a variable (data item) has been observed to occur in relation to the total number of values for that variable.
The relative frequency is calculated by dividing the absolute frequency by the total number of values for the variable.
How are relative frequencies expressed?
Ratios, rates, proportions and percentages are different ways of expressing relative frequencies.
A ratio compares the frequency of one value for a variable with another value for the variable.
The first value identified in a ratio must be to the left of the colon (:) and the second value must be to the right of the colon (1st value : 2nd value).
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the ratio of heads to tails is 12:8. Alternatively, the ratio of tails to heads is 8:12.
A rate is a measurement of one value for a variable in relation to another measured quantity.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the rate is 12 heads per 20 coin tosses. Alternatively, the rate is 8 tails per 20 coin tosses.
A proportion describes the share of one value for a variable in relation to a whole.
It is calculated by dividing the number of times a particular value for a variable has been observed, by the total number of values in the population.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the proportion of heads is 0.6 (12 divided by 20). Alternatively, the proportion of tails is 0.4 (8 divided by 20).
A percentage expresses a value for a variable in relation to a whole population as a fraction of one hundred.
The percentage total of an entire dataset should always add up to 100, as 100% represents the total, it is equal to the ‘whole’. A percentage is calculated by dividing the number of times a particular value for a variable has been observed, by the total number of observations in the population, then multiplying this number by 100.
For example, in a total of 20 coin tosses where there are 12 heads and 8 tails, the percentage of heads is 60% (12 divided by 20, multiplied by 100). Alternatively, the percentage of tails is 40% (8 divided by 20, multiplied by 100).
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In the construction of a histogram, there are several steps that we must undertake before we actually draw our graph. After setting up the classes that we will use, we assign each of our data values to one of these classes then count the number of data values that fall into each class and draw the heights of the bars. These heights can be determined by two different ways that are interrelated: frequency or relative frequency.
The frequency of a class is the count of how many data values fall into a certain class wherein classes with greater frequencies have higher bars and classes with lesser frequencies have lower bars. On the other hand, relative frequency requires one additional step as it is the measure of what proportion or percent of the data values fall into a particular class.
A straightforward calculation determines the relative frequency from the frequency by adding up all the classes' frequencies and dividing the count by each class by the sum of these frequencies.
To see the difference between frequency and relative frequency we will consider the following example. Suppose we are looking at the history grades of students in 10th grade and have the classes corresponding to letter grades: A, B, C, D, F. The number of each of these grades gives us a frequency for each class:
- 7 students with an F
- 9 students with a D
- 18 students with a C
- 12 students with a B
- 4 students with an A
To determine the relative frequency for each class we first add the total number of data points: 7 + 9 + 18 + 12 + 4 = 50. Next we, divide each frequency by this sum 50.
- 0.14 = 14% students with an F
- 0.18 = 18% students with a D
- 0.36 = 36% students with a C
- 0.24 = 24% students with a B
- 0.08 = 8% students with an A
The initial data set above with the number of students who fall into each class (letter grade) would be indicative of the frequency while the percentage in the second data set represents the relative frequency of these grades.
An easy way to define the difference between frequency and relative frequency is that frequency relies on the actual values of each class in a statistical data set while relative frequency compares these individual values to the overall totals of all classes concerned in a data set.
Either frequencies or relative frequencies can be used for a histogram. Although the numbers along the vertical axis will be different, the overall shape of the histogram will remain unchanged. This is because the heights relative to each other are the same whether we are using frequencies or relative frequencies.
Relative frequency histograms are important because the heights can be interpreted as probabilities. These probability histograms provide a graphical display of a probability distribution, which can be used to determine the likelihood of certain results to occur within a given population.
Histograms are useful tools to quickly observe trends in populations in order for statisticians, lawmakers, and community organizers alike to be able to determine the best course of action to affect the most people in a given population.