Introduction to Cumulative Frequency Distribution
We come across the term “frequency” in statistics. Collecting the data, presenting the data in a particular format and analyzing the data is called Statistics. This data collected can be analyzed and represented through various means like graphs, tables, pie charts, histograms etc. In simple terms, frequency is the number of times a number or value repeats in the given data. A frequency distribution table in statistics tell us comprehensively what the different variables present in the entire data and how many times they occur i.e, frequency of the variables.
What is Cumulative Frequency Distribution?
The distribution of any data using a table or graph makes the data comprehensive. There are different classes and subclasses, which indicates the frequency, and this is known as the frequency distribution of data. The frequency is actually the number of times an observation occurs for a particular time period.
Cumulative frequency helps to determine the number of operations, lying above specific observation. For calculating the increasing frequency of any data, the frequency of the observation is added to the sum of frequencies of the predecessors. This previous sum is called the cumulative frequency distribution of the entire set of data.
How to Find Cumulative Frequency and Why to Use Graphs to Represent it
While graphical representation makes the data simpler to understand, one can even see the fluctuations and ups and downs easily by drawing graphs. Usually, bar graphs and frequency polygons are used for graphical representation of the frequencies. The graphical representation of cumulative incidence is called ogive.
The graph for the frequencies can be plotted in two different ways.
Cumulative frequency distribution curve, less than type
Cumulative frequency distribution curve, more than type
The below-given solution will provide a stepwise understanding of how to find more than the cumulative distribution frequency and how to calculate less than type cumulative distribution frequency.
How to Calculate Cumulative Frequency Distribution
How to Calculate More Than Cumulative Frequency
In the above-given table, four different groups are representing the daily wage of workers, along with the number of workers in the company. For example, when looking at the first group in the table, there are ten workers with regular income between 100-120 rupees.
Step 1: The number of worker columns is the frequency column. Frequency is nothing but the rate at which activity has occurred, which is further denoted by F. The sum of all the frequencies is 50. Cumulative frequency is the running frequency of every group present in the table.
Step 2: As we are solving more than one type, it's essential to take the lower limits from the table, which is 100,120,140 and 160. If we calculate the lower bound from the table, it will give a result like more than 100, more than 120, and so on.
Step 3: As the total frequency, we have obtained 50; the very first cumulative frequency will be written as 50. The cumulative frequency of the second row will be more than 120 and will be under 180. Hence, we have to subtract ten from 50 and progressively subtract the frequency from the resultant cumulative frequency. In more than cumulative type frequency, the last frequency should match with the cumulative frequency.
Plotting the Frequency Distribution Graph
To draw the cumulative frequency distribution graph, the following steps have to be taken care of:
As the cumulative frequency is a dependent variable, it will come on the Y-axis, and the daily income will be shown on the X-axis. If there's a discontinuity of the numbers on the x-axis, then a key should be made before all the numbers. Don't forget to set the scale before jotting the graph. Start plotting the points with the help of numbers given on the table.
After joining all the points through the freehand, the frequency distribution curve will be a decreasing curve, because in more than one type the curve which will be obtained will always be a decreasing curve.
Graph for Cumulative Frequency Distribution More than Type
How to Calculate Less than Cumulative Frequency?
The above-given example is similar to the more than type, but there are some changes made. The class size of every group is 20. Over here, we will be using the upper limits and not the lower limits.
Step 1 and Step 2 are the same as the more than type frequency. Hence we will straight away move to step 3.
Step 3: The frequency and the cumulative frequency for the first group will be the same, which is 12. For the second group, one has to add the frequencies between 100-140, that's 10+12, 22. In the end, the cumulative incidence obtained will be 50. The total frequency will always be equal to the last cumulative frequency.
Plotting the Frequency Distribution Graph
To draw a cumulative frequency distribution graph, The X-axis will be of daily income, in rupees and the Y-axis will be of cumulative frequency. After setting the scale, start to plot the points on the graph. After making a freehand curve, the frequency distribution curve will be an increasing curve, unlike the more than type graphs.
Calculating the median is also of less than cumulative type frequency is as same as more than type frequency graph.
Note: A sigma (Σ) denotes frequency.
Importance of Learning Cumulative Frequency in Statistics
Cumulative frequency is one of the important features of statistics. It is used to represent the raw data better and understand the data comprehensively and easily. It helps in a proper detailed study in a field study, as it is a positivistic tool, the chance of error is less.
Quantitative data which we exactly need is collected by statistics and it improves our research. Many daily life events like weather forecasts, census, thermal graphs are dependent on statistics and related concepts. Even the samples collected to analyze data related to pandemics are based on statistical techniques like cumulative frequency.
FAQs on Cumulative Frequency Distribution
1. The Cumulative Frequency Distribution is Shown By
Graphs can represent the cumulative frequency in statistics. All the numbers have a specific rate, and with this data, a chart is made. A frequency polygon does the graphical representation of the data. Usually, a histogram is provided, and one has to find the cumulative frequency of that histogram. Technically, the aggregate amount obtained by you at the end is summarised by all the other numbers provided by you. Although there are two ways to find the cumulative frequency, there is just the slightest difference in the calculation part, like in less than type the first frequency and cf will be the same, but in more than type, the cf will be the sum of all the frequencies.
2. How to Construct Cumulative Frequency Distribution
There will be a table and a frequency column provided to you already. Calculate the cf, if it is more than type, then the first cf will be the sums of all the frequencies, and all the further will be subtracted progressively by previous frequencies. And for less than type, the first frequency and cf will be the same, and you have to add all the further frequencies progressively. It's not mandatory to draw a graph, but a graphical representation of the data not only helps to show the trend of that particular question but also it helps to find the mean and the median in a distinctive way.
3. What is a cumulative frequency?
Cumulative frequency is the sum of all the frequencies until we determine the data according to a class interval. The total from the very first frequency to the last frequency is cumulative frequency, it helps in dividing the entire data into intervals. It can be derived from adding the frequency of the first interval to the frequency of the second interval and adding it up to the final interval to derive the cumulative frequency.
A table formed out of such frequency intervals is called a cumulative frequency table. It further has various classes and subclasses. To explain simply, the frequency of a number is the number of times it occurs in a data or sequence. This cumulative frequency helps us measure the operations that are above the particular observation. This can be measured manually or can be plotted on a graph.
4. What are the types of cumulative frequency?
Following are the types of cumulative frequency:
Lesser than frequency is a type of cumulative frequency that is derived by adding all the frequencies to the previous class intervals and also to the interval in which it exists.
Greater than frequency is a type of cumulative frequency derived from adding all the frequencies from the highest frequency to the lowest frequency. Greater than cumulative frequency is also known as more than a cumulative type of frequency.
5. Explain how relative frequency is different from cumulative frequency?
Frequency, explained simply, is the total number of times a particular value occurs in the entire data. Relative frequency can be expressed in decimals and tells us the relative times a value occurs in the data. Cumulative frequency, on the other hand, is the sum of all the frequencies from the first till the ending to derive a class interval. It is done by adding the frequency of the first class interval to the second class interval and both of them to the frequency of the third class interval and so on till the end.