How To Draw A Histogram For Grouped Data
When dealing with enormous amounts of data, information grouping is quite essential. A pictogram or a bar graph can alternatively be used to illustrate this grouped data. Grouped data is data created by individual grouping observations of a variable into groups. A frequency distribution tabular array of these groups may be used to summarise or analyse data. A histogram is a graph that displays frequencies for intervals of values of a metric variable. A histogram is similar to a bar graph. Withal, it is used for class intervals that are not interrupted. It also displays a set of continuous data'south underlying frequency distribution. This article discusses grouping information and histograms in particular. Read on to acquire more.
Group Data
When the number of observations is considerable, we can use the grouping of data idea to separate the data into several categories. Private observations of a variable are grouped into groups, and the frequency distribution tabular array of these groups is a helpful way to summarise the data.
Study The Concept Of Ungrouped Information
Frequency Distribution
The data gathered is organised in a tabular array using a frequency distribution. Students' grades, weather in other cities, lucifer points, and and so on might all be incorporated in the data. Following the collection of data, we must present it in a meaningful manner to aid comprehension. Arrange the data in a table such that all of its features are summarised.
Let us consider an example. The following are the temperature of \(20\) cities in March in degrees.
\(30,\,25,\,27,\,twenty,\,25,\,xxx,\,20,\,27,\,24,\,24,\,25,\,27,\,twenty,\,30,\,33,\,25,\,33,\,33,\,27,\,20\). Permit us present this data in a table and make up one's mind the frequency of the cities with the same temperature.
| Temperature | Number of Cities |
| \(twenty\) | \(4\) |
| \(24\) | \(2\) |
| \(25\) | \(4\) |
| \(27\) | \(4\) |
| \(thirty\) | \(three\) |
| \(33\) | \(iii\) |
All the collected data is arranged under the temperature column and the number of cities columns, as tin be seen. This organization makes it simple to empathise the information provided, and we tin see the number of cities with the aforementioned temperature.
As a result, frequency distribution in statistics aids u.s. in organising data in such a way that its features may be easily understood at a glance.
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Steps to Depict Frequency Distribution Tabular array for Grouped Data
We will follow the below steps to depict a frequency distribution table for the grouped data.
- Divide the data into groups using the data provided.
- Sort the observations into ascending order.
- Find out the frequency of each ascertainment.
- In the frequency distribution table, write the frequency and group proper noun.
Advantages of Grouping Data
Below are some of the advantages of grouping information.
- Group information aids in focusing on crucial subpopulations while ignoring others that aren't.
- Data grouping enhances estimating accurateness and efficiency.
Ungrouped Data
Raw data, also known as ungrouped information, is data nerveless from direct observation.
- Example: Consider the marks of \(xx\) students of a class in a particular examination.
- \(xl,\,50,\,50,\,56,\,92,\,60,\,70,\,60,\,88,\,76\)
- \(88,\,80,\,70,\,72,\,92,\,36,\,forty,\,forty,\,70,\,36\)
In the higher up example, the number of students who obtained the same number of marks is chosen the frequency. Here, \(3\) students got \(forty\) marks. So, the frequency of \(40\) is \(3\).
Histogram
A histogram is a graphical depiction of a fix of data that is divided into user-defined ranges. Like a bar graph, the histogram turns a information serial into an easily understandable visual by group data points into logical ranges or bins.
The histogram is made upwardly of a series of bars (similar to a bar nautical chart), merely these bars are next to one some other, and the height of the bars is proportional to the frequency of the diverse classes. The expanse of each rectangle denoted the frequency of each class.
The rectangles all accept the aforementioned width, and their heights direct lucifer the class frequencies when the class intervals are equal.
If the length of the appropriate class interval rises, the height of a rectangle must exist proportionally reduced.
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Histogram for a Grouped Information
A histogram is a 2-dimensional graphical delineation of a continuous frequency distribution.
The bars in a histogram are e'er put side by side, with no gaps between them. That is, rectangles are built on the distribution'due south class intervals in histograms. The frequencies are proportional to the rectangle areas.
Let us now examine the procedures involved in creating a histogram for grouped data.
- If the data is in a discontinuous form, stand for information technology in a continuous class.
- On a compatible calibration, marking the class intervals along the \(x-\)axis.
- On a consequent calibration, mark the frequencies forth the \(y-\)centrality.
- Create rectangles with grade intervals equally the bases and frequencies every bit the heights.
Histogram for Ungrouped Data
The histogram is created by plotting the form boundaries (not class limits) on the \(x-\)axis and the respective frequencies on the \(y-\)axis from the grouped information. Earlier constructing a histogram with ungrouped data, nosotros must first create a grouped frequency distribution.
Bar graphs are often used for discrete and categorical data. However, in rare cases where an approximation is required, a histogram may exist generated. The steps for creating a histogram for ungrouped data are equally follows:
- Mark the possible values on \(x-\)axis.
- Mark the frequencies along the \(y-\)centrality.
- Depict a rectangle centred on each value, with equal width on each side and a margin of \(0.5\) on either side.
Solved Examples on Group Data and Histogram
Q.one. Depict a histogram for the below table, which represent the marks obtained by \(100\) students in an test:
| Marks | Number of Students |
| \(0 – 10\) | \(5\) |
| \(10 – twenty\) | \(x\) |
| \(xx – 30\) | \(xv\) |
| \(30 – 40\) | \(20\) |
| \(forty – fifty\) | \(25\) |
| \(50 – threescore\) | \(12\) |
| \(60 – 70\) | \(viii\) |
| \(70 – 80\) | \(v\) |
Ans: The class intervals are all the same length, at ten marks each. Let's draw a line on the \(x-\)axis to correspond these course intervals. Along the \(y-\)centrality, write the number of students on the appropriate scale. Below is a representation of the histogram.
Calibration: \(ten-\)axis: \(1\;\rm{cm} = ten\) marks
\(y-\)centrality: \(1\;\rm{cm} = v\) students
The bars in the diagram to a higher place are drawn in a continuous pattern. The rectangles have lengths (heights) that are proportionate to the frequencies. The areas of the confined are proportional to the respective frequencies because the grade intervals are equal.
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Q.2. Depict a histogram to represent the beneath data :
| Class Interval | Frequency |
| \(0 – ten\) | \(8\) |
| \(ten – 20\) | \(12\) |
| \(20 – xxx\) | \(6\) |
| \(30 – 40\) | \(14\) |
| \(40 – 50\) | \(10\) |
| \(50 – sixty\) | \(five\) |
Ans: The histogram for the given data is fatigued below.
Scale: \(x-\)axis: \(ane\;\rm{cm} = 10\) units
\(y-\)axis: \(ane\;\rm{cm} = 2\) units
Q.3. A teacher wanted to analyse the performance of 2 sections of students in a mathematics examination of \(100\) marks. Looking at their performances, she establish that a few students got under \(xx\) marks and a few got \(70\) marks or to a higher place. And so she decided to group them into intervals of varying sizes: \(0 – 20,\,xx – 30,….\;threescore – 70,\,70 – 100\). Then she formed the following table:
| Marks | Number of Students |
| \(0 – 20\) | \(7\) |
| \(twenty – thirty\) | \(10\) |
| \(xxx – forty\) | \(10\) |
| \(40 – l\) | \(twenty\) |
| \(50 – 60\) | \(20\) |
| \(60 – 70\) | \(15\) |
| \(70 -\)above | \(viii\) |
| Total | \(xc\) |
Ans: We need to make specific changes in the lengths of the rectangles so that the areas are again proportional to the frequencies.
The steps to be followed are given below:
i. Select the class interval with the minimum course size.
2. The lengths of the rectangles are so modified to be proportionate to the class size.
When the class size is \(twenty\), the length of the rectangle is \(7\). Then, when the class size is \(10\), the length of the rectangle volition exist \(\frac{7}{{xx}} \times ten = 3.five\)
Therefore, the modified table will exist as follows.
| Marks | Frequency | Width of the form | Length of the rectangle |
| \(0 – xx\) | \(7\) | \(20\) | \(\frac {vii}{20} \times ten = 3.5\) |
| \(20 – 30\) | \(x\) | \(x\) | \(\frac {x}{ten} \times 10 = 10\) |
| \(30 – xl\) | \(10\) | \(10\) | \(\frac {10}{ten} \times 10 = x\) |
| \(40 – 50\) | \(20\) | \(x\) | \(\frac {twenty}{10} \times 10 = twenty\) |
| \(50 – sixty\) | \(20\) | \(10\) | \(\frac {20}{ten} \times x = 20\) |
| \(60 – 70\) | \(xv\) | \(10\) | \(\frac {15}{ten} \times ten = fifteen\) |
| \(70 – 100\) | \(8\) | \(30\) | \(\frac {8}{30} \times 10 = 2.67\) |
Since we accept calculated these lengths for an interval of \(10\) marks in each case, we may telephone call these lengths as "proportion of students per \(10\) marks interval".
So, the right histogram with varying width is given in the below figure.
Q.four. In a report of covid patients in a village, the following observations were noted. Stand for the given data by using the histogram.
| Ages | Number of patients |
| \(ten – 20\) | \(three\) |
| \(twenty – 30\) | \(half dozen\) |
| \(30 – 40\) | \(thirteen\) |
| \(40 – 50\) | \(20\) |
| \(50 – lx\) | \(10\) |
| \(sixty – 70\) | \(5\) |
Ans: The histogram for the given data is drawn beneath.
Scale: \(10-\)axis: \(ane\;\rm{cm} = 10\) age
\(y-\)axis: \(1\;\rm{cm} = 2\) patients
Q.v. Draw the histogram for the beneath information.
| Groups | Frequency |
| \(0 – x\) | \(3\) |
| \(10 – 20\) | \(11\) |
| \(20 – thirty\) | \(14\) |
| \(30 – 40\) | \(14\) |
| \(40 – 50\) | \(eight\) |
Ans: The histogram for the given data is fatigued below.
Scale: \(x-\)axis: \(one\;\rm{cm} = 10\) units
\(y-\)axis: \(1\;\rm{cm} = five\) units
Summary
In this article, nosotros learnt the definitions of grouping data, frequency distribution, ungrouped data, and histograms. As well, we have studied the method to describe histograms for grouped and ungrouped data and solved some example issues on the same. We tin group large quantities of data by making frequency distribution tables for to represent the frequency of occurrence of each data value. For fifty-fifty larger data sets, we tin can make grouped frequency distribution tables for a set of data values. In a histogram the height of the bars represents the frequency of the class-interval and the horizontal centrality represents the data groups. There is no gap between the bars in a histogram.
FAQs on Grouping Data and Histogram
The answers to the near ordinarily asked questions on Grouping Data and Histogram are provided here:
Q.1. What practise you lot mean by the grouping of data?
Ans: When the number of observations is considerable, nosotros tin use the grouping of data idea to split up the data into several categories. Individual observations of a variable are grouped into groups, and the frequency distribution tabular array of these groups is a helpful way to summarise the data.
Q.two. What is group data and histogram?
Ans: When the number of observations is considerable, we can use the grouping of data thought to split the information into several categories. Individual observations of a variable are grouped into groups, and the frequency distribution table of these groups is a helpful way to summarise the information.
A histogram is a graphical depiction of a gear up of data that is divided into user-divers ranges. Like a bar graph, the histogram turns a information series into an easily understandable visual by grouping information points into logical ranges or bins.
Q.iii. What is the purpose of grouping data?
Ans: Grouping data aids in focusing on crucial subpopulations while ignoring others that aren't. Information grouping enhances estimating accuracy and efficiency.
Q.iv. How practice y'all group data in statistics?
Ans: The post-obit steps in the grouping can be summarised:
1. Determine the number of classes.
2. Summate the range or the divergence betwixt the data's highest and lowest observations.
3. To estimate the size of the interval, split the range by the number of classes.
iv. To calculate the upper-form limit, have the lower course limit of the lowest class and multiply it by the class interval.
Q.five. What is an ungrouped data case?
Ans: Raw data, also known equally ungrouped information, is information nerveless from direct observation.
Case: Consider the marks of \(20\) students of a class in a particular exam.
\(40,\,50,\,50,\,56,\,92,\,lx,\,seventy,\,60,\,88,\,76\)
\(88,\,fourscore,\,70,\,72,\,92,\,36,\,40,\,40,\,70,\,36.\)
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At present that you have a detailed commodity on Grouping Data and Histogram, we hope yous report well. If you get stuck somewhere, practise permit usa know in the comments sections. We will go dorsum to you at the earliest.
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