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Frequency Distribution
Introduction
It is sometimes important to know how much data or a data set has repeated itself in a given number of functions that include the data. Such inferences help the examiner to identify whether some data sets are predominant or whether some data has repeated itself for lower times than expected. For such inferences, frequency distribution comes in handy.
Frequency Distribution: Definition
Frequency distribution gives the number of observations of data in tabular or graphical form. The representation of observations is done within specified intervals in the frequency distribution. Therefore, frequency distribution must have specified intervals that are evenly distributed.
Frequency distributions are part of statistics and they are particularly useful for normal distributions. The use of frequency distributions for normal distributions shows the observation of various probabilities divided among standard deviations.
Frequency distribution often offers a visual representation of the distribution of observations. So, it helps the users of data understand the nature of the distributions of observations. Data collected as samples often need to be represented in visual forms. This is done to make data more comprehensible and useful. Therefore, when used in its true form, frequency distribution can help analysts make better inferences about the sample.
For example, in comparing the income of managers, there is a high probability that some get more wages and some get lower than the average, but there is a high probability that the concentration of middle-range income is higher than all others. A frequency distribution can help analysts get a good idea of this by looking at the distribution which is done via a graph or a table of frequencies.
The Visual Representation
Usually, the visual representation of frequency distribution is done using histograms and bar charts. In such representations, the y-axis represents the frequencies while the x-axis represents the variables to be measured.
For example, in the wages of managers, the y-axis is the number of managers while the x-axis will present the wages.
Histogram charts usually represent normal distributions. This means that most of the frequencies will gather in the middle columns. These normal distributions can show probabilities of observations that are divided among standard deviations.
Pie charts are also used for frequency distribution where the percentage of frequencies is used as sections of groups. It shows the particular parts of data out of the total data considered in a circular diagram.
Frequency Polygon: When mid-points of histograms are joined with straight lines a frequency polygon is obtained.
Types of Frequency Distributions
There are four types of frequency distributions and two types of frequency distribution tables in statistics. They are –
- Ungrouped Frequency Distribution
- Grouped Frequency Distribution
- Relative Frequency Distribution
- Cumulative Frequency Distribution
Ungrouped Frequency Distribution
In an ungrouped frequency distribution, the frequencies are distributed singularly. There is no group of data values in an ungrouped frequency distribution.
Grouped Frequency Distribution
Unlike ungrouped frequency distribution, in a grouped frequency distribution, the data is divided and collected under groups of frequencies. These groups are known as intervals. Therefore, grouped frequency distribution shows frequency distributions in terms of class intervals.
Relative Frequency Distribution
It shows the proportions of frequency distribution to the total number of distribution observations in each category.
Cumulative Frequency Distribution
It is the addition of the first frequency and all other frequencies available in a frequency distribution. In this type of frequency distribution, one has to take the next frequency and add a value to it.
After the addition, the sum value of the addition is added to the next value and so on until the last value is obtained. The last value obtained this way will be the cumulative frequency of the distribution.
Frequency Distribution Tables
Frequency distribution tables are tabular forms of frequency distribution.
To make the frequency distribution table, each data is collected and then placed in the table in groups or non-grouped tables. The tally marking system is a way to create a tabular form of distribution tables. In the tally marking system, five data are taken to form a set and each value is placed in the frequency portion of each data set. Once five data form a set, one tally set is completed and it is crossed with the fifth data.
This way, all the data are collected in a table and finally calculated for each attribute. The tally marking system is an easy and simple way to collect individual raw data and convert them into a comprehensible form.
Types of Frequency Distribution Table
Following are the types of frequency distribution table:
Grouped Frequency Distribution Table
It is used for large numbers of data. Class intervals are formed with equal intervals and data is put in the groups where they belong one after another. Finally, the tabular form of available data is obtained in this process.
An example of grouped frequency distribution table is as follows.
Suppose out of fifty children in a class, ten have got marks above eighty. Their marks are as follows: 81, 82, 86, 83, 92, 94, 86, 87, 85, 90. This can be shown in tabular form as follows:
| Marks | Students |
|---|---|
| 80–85 | 4 |
| 86–90 | 4 |
| 91–95 | 2 |
| 96–100 | 0 |
Ungrouped Frequency Distribution Table
In such types of table, there is no grouping depending on the nature of data. Individual data is placed in the table against which the frequencies are put in the table of this type of frequency distribution table.
An example like the above would be a number of students getting particular marks in an exam.
| Marks | Students |
|---|---|
| 85 | 4 |
| 86 | 2 |
| 90 | 1 |
| 93 | 3 |
Grouped and ungrouped frequency distribution both offer planned insight into the raw data. However, grouped data is considered superior because it is easy to infer and use in comparison to ungrouped distribution. However, they have their own advantages and disadvantages depending upon which particular tables can be prepared. It, therefore, depends on the user which form of a table to make for easier inferences and actionable insight into the data.
It is notable that taking too long or too short intervals can lead to errors in the study. Therefore, one must choose the intervals optimally that are relatable to the purpose of the study.
Conclusion
Frequency distribution is a popular method in statistics to provide data with an insightful shape. Classification and grouping of data in various forms are useful for researchers in various ways. For example, frequency distribution can be used to offer ideas about the various data obtained during statistical and scientific studies.
To offer a ready solution and useful data to researchers, the frequency distribution is an important tool. Although it is not always easy to stratify data, frequency distributions offer a way to bring uniformity to data collection and management. For this quality, the frequency distribution is unavoidable.
FAQs
Q1. What are the two types of frequency distribution tables and what is the biggest difference between them?
Ans. The two types of frequency distribution tables are grouped and non-grouped frequency distribution tables. The biggest difference between them is that grouped frequency distribution table has intervals while the non-grouped one has no intervals and works on individual forms of data. The grouped frequency distribution is, therefore, continuous in nature.
Q2. What is meant by frequency?
Ans. Frequency is the number of occurrences of a particular item. It is often expressed in terms of individual numbers.
Q3. How is frequency distribution presented?
Ans. Frequency distributions are presented through graphs or data sets.
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- Find the mean, mode and median of the following frequency distribution:
- Find the missing frequency $(p)$ for the following distribution whose mean is 7.68.
- Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50.
- The mean of the following distribution is 18. Find the frequency f of the class 19-21.
- Given below is a cumulative frequency distribution table showing the ages of people living in a locality.
- Construct a frequency distribution table for the following weights (in grams) of 35 mangoes, using the equal class intervals, one of them is \( 40-45(45 \) not included).\( 30,40,45,32,43,50,55,62,70,70,61,62,53,52,50,42,35,37,53,55,65,70,73,74,45,46,58 \),\( 59,60,62,74,34,35,70,68 \)(a) How many classes are there in the frequency distribution table?(b) Which weight group has the highest frequency?
- Difference between Selective Distribution and Exclusive Distribution
- Find the missing frequency $p$ for the following distribution whose mean is 7.68.$x$:35791113$f$:6815$p$84.
- Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50.$x$:1030507090$f$:17$f_1$32$f_2$19Total 120.
- The following distribution gives the daily income of 50 workers of a factory:Daily income (in Rs):100-120120-140140-160160-180180-200Number of workers:12148610Convert the above distribution to a less than type cumulative frequency distribution and draw its ogive.
- The following distribution gives daily income of 50 workers of a factory:Daily income (in Rs):200-220220-240240-260260-280280-300Number of workers:12148610Convert the distribution above to a 'less than type' cumulative frequency distribution and draw its ogive.
