Cumulative Frequency | Mathematics for Grade 10 PDF Download

Introduction

Cumulative frequency is the total of a frequency and all frequencies in a frequency distribution until a certain defined class interval. The running total of frequencies starting from the first frequency till the end frequency is the cumulative frequency. The total and the data are shown in the form of a table where the frequencies are divided according to class intervals. Let us learn more about cumulative frequency, plotting a cumulative frequency graph, and learn to read a cumulative frequency table along with solving examples.

Definition of Cumulative Frequency

In statistics, the frequency of the first-class interval is added to the frequency of the second class, and this sum is added to the third class and so on then, frequencies that are obtained this way are known as cumulative frequency (c.f.). A table that displays the cumulative frequencies that are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table. There are two types of cumulative frequency - lesser than type and greater than type. Cumulative frequency is used to know the number of observations that lie above (or below) a particular frequency in a given data set. Let us look at a few examples that are used in many real-world situations.

Example 1: Robert is the sales manager of a toy company. On checking his quarterly sales record, he can observe that by the month of April, a total of 83 toy cars were sold.

Note how the last cumulative total will always be equal to the total for all observations since all frequencies will already have been added to the previous total. Here,83=20+30+15+18

Example 2: A Major League Baseball team records its home runs in the 2020 session as given below.

From the above table, it can be observed that the team made 29 home runs before playing in the finals.

Types of Cumulative Frequency

Cumulative frequency is the total frequencies showcased in the form of a table distributed in class intervals. There are two types of cumulative frequency i.e. lesser than and greater than, let us learn more about both types.

Lesser Than Cumulative Frequency
Lesser than cumulative frequency is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The cumulate starts from the lowest to the highest size. In other words, when the number of observations is less than the upper boundary of a class that's when it is called lesser than cumulative frequency.

Greater Than Cumulative Frequency
Greater than cumulative frequency is obtained by finding the cumulative total of frequencies starting from the highest to the lowest class. It is also called more than type cumulative frequency. In other words, when the number of observations is more than or equal to the lower boundary of the class that's when it is called greater than cumulative frequency.
Let us look at example to understand the two types.

Example: Write down less than type cumulative frequency and greater than type cumulative frequency for the following data.
Height (in cm) Frequency (students)
140 – 145 2
145 – 150 5
150 – 155 3
155 – 160 4
160 – 165 1

We would have less than type and more than type frequencies as:
The following information can be gained from either graph or table

• Out of a total of 15 students, 8 students have a height of more than 150 cm
• None of the students are taller than 165 cm
• Only one of the 15 students has a height of more than 160 cm

Constructing a Cumulative Frequency Distribution Table

A cumulative frequency table is a simple visual representation of the cumulative frequencies for different values or categories. To construct a cumulative frequency distribution table, there are a few steps that can be followed which makes it simple to construct. Let us see what the steps are:

• Step 1: Use the continuous variables to set up a frequency distribution table using a suitable class length.
• Step 2: Find the frequency for each class interval.
• Step 3: Locate the endpoint for each class interval (upper limit or lower limit).
• Step 4: Calculate the cumulative frequency by adding the numbers in the frequency column.
• Step 5: Record all results in the table.

Example: During a 20-day long skiing competition, the snow depth at Snow Mountain was measured (to the nearest cm) for each of the 20 days. The records are as follows: 301, 312, 319, 354, 359, 345, 348, 341, 347, 344, 349, 350, 325,323, 324, 328,322, 332, 334, 337.

Given measurements of snow depths are: 301, 312, 319, 354, 359, 345, 348, 341, 347, 344, 349, 350, 325,323, 324, 328,322, 332, 334, 337

• Step 1: The snow depth measurements range from 301 cm to 359 cm. To produce the frequency distribution table, the data can be grouped in class intervals of 10 cm each.
  In the Snow depth column, each 10-cm class interval from 300 cm to 360 cm is listed.
• Step 2: The frequency column will record the number of observations that fall within a particular interval. The tally column will represent the observations only in numerical form.
• Step 3: The endpoint is the highest number in the interval, regardless of the actual value of each observation.
  For example, in the class interval of 311-320, the actual value of the two observations is 312 and 319. But, instead of using 219, the endpoint of 320 is used.
• Step 4: The cumulative frequency column lists the total of each frequency added to its predecessor.

Using the same steps mentioned above, a cumulative frequency distribution table can be made as:

Constructing Cumulative Frequency Distribution Graph

The cumulative frequency distribution of grouped data can be represented on a graph. Such a representative graph is called a cumulative frequency curve or an ogive. Representing cumulative frequency data on a graph is the most efficient way to understand the data and derive results. In the world of statistics, graphs, in particular, are very important, as they help us to visualize the data and understand it better. So let us learn about the graphical representation of the cumulative frequency. There are two types of Cumulative Frequency Curves (or Ogives): More than type Cumulative Frequency Curve and Less than type Cumulative Frequency Curve.

More Than Cumulative Frequency Curve

In the more than cumulative frequency curve or ogive, we use the lower limit of the class to plot a curve on the graph. The curve or ogive is constructed by subtracting the total from first-class frequency, then the second class frequency, and so on. The upward cumulation result is more than or greater than the cumulative curve. The steps to plot a more than curve or ogive are:

• Step 1: Mark the lower limit on the x-axis
• Step 2: Mark the cumulative frequency on the y-axis.
• Step 3: Plot the points (x,y) using lower limits (x) and their corresponding Cumulative frequency (y).
• Step 4: Join the points by a smooth freehand curve.

Less Than Cumulative Frequency Curve

In the mess than cumulative frequency curve or ogive, we use the upper limit of the class to plot a curve on the graph. The curve or ogive is constructed by adding the first-class frequency to the second class frequency to the third class frequency, and so on. The downward cumulation result is less than the cumulative frequency curve. The steps to plot a less than cumulative frequency curve or ogive are:

• Step 1: Mark the upper limit on the x-axis
• Step 2: Mark the cumulative frequency on the y-axis.
• Step 3: Plot the points (x,y) using upper limits (x) and their corresponding Cumulative frequency (y).
• Step 4: Join the points by a smooth freehand curve.

Example: Graph the two ogives for the following frequency distribution of the weekly wages of the given number of workers.

Less than curve or ogive:
Mark the upper limits of class intervals on the x-axis and take the less than type cumulative frequencies on the y-axis. For plotting less than type curve, points (20,4), (40,9), (60,15), and (80,18) are plotted on the graph and these are joined by freehand to obtain the less than ogive.

Greater than curve or ogive:
Mark the lower limits of class intervals on the x-axis and take the greater than type cumulative frequencies on the y-axis. For plotting greater than type curve, points (0,18), (20,14), (40,9), and (60,3) are plotted on the graph and these are joined by freehand to obtain the greater than type ogive.
A perpendicular line on the x-axis is drawn from the point of intersection of these curves. This perpendicular line meets the x-axis at a certain point, this determines the median. Here the median is 40. The median of the given data could also be found from cumulative graphs. On drawing both the curves on the same graph, the point at which they intersect, the corresponding value on the x-axis, represents the median of the given data set.
The less than and greater than ogives shown in the graph below.

Relative Cumulative Frequency Graph

Relative cumulative frequency graphs are a type of ogive graphs that showcases the percentile of the given data. The ogive shows at what percent of the data is below a particular value. In other words, relative cumulative frequency graphs are ogive graphs that show the cumulative percent of the data from left to right. The two main aspects of this type of graph are, it shows the percentile and indicates the shape of the distribution. Percentiles is the data that is either in the ascending or descending order into 100 equal parts. It indicates the percentage of observations a value is above. Whereas a shape of the distribution helps in transforming observations using standard deviations to see how far specific observations are from the mean. One observation can be compared to another by standardizing the dataset. This particular aspect is widely used in statistics. Let us look at an example:

Example: A car dealer wants to calculate the total sales for the past month and wants to know the monthly sales in percentage after weeks 1, 2, 3, and 4. Create a relative cumulative frequency table and present the information that the dealer needs.

First total up the sales for the entire month:
10 + 17 + 14 + 11 = 52 cars
Then find the relative frequencies for each week by dividing the number of cars sold that week by the total:

• The relative frequency for the first week is: 10/52 = 0.19
• The relative frequency for the second week is: 17/52 = 0.33
• The relative frequency for the third week is: 14/52 = 0.27
• The relative frequency for the fourth week is: 11/52 = 0.21

To find the relative cumulative frequencies, start with the frequency for week 1, and for each successive week, total all of the previous frequencies
Note: The first relative cumulative frequency is always the same as the first relative frequency, and the last relative cumulative frequency is always equal to 1.

Solved Example

Example 1: Create a cumulative frequency table showing the number of hours per week that Ryan plays video games, based on the given information.
Ryan's Game Time
Monday: 2 hrs
Tuesday: 1 hr
Wednesday: 2 hrs
Thursday: 3 hrs
Friday: 4 hrs
Saturday: 2 hrs
Sunday: 1 hr

A cumulative frequency table for Ryan's game time can be made as follows:
Thus, Ryan spends 15 hours of gaming in a week.

Example 2: A weather forecaster highlights the lows over-night for the past 10 days in a small town in Wisconsin. The temperature readings are given in degrees Fahrenheit and are shown below. Use the data to make a frequency table. 41, 58, 41, 54, 49, 46, 52, 53, 55, 52

Frequency is nothing but the number of times an event occurs in a given scenario.
We will first choose a suitable class interval for the above data, then we will enter the frequency values to complete the table.

Example 3:The following represents scores that a class of 20 students received on their most recent Biology test. Plot a less than type Ogive.
58, 79, 81, 99, 68, 92, 76, 84, 53, 57, 81, 91, 77, 50, 65, 57, 51, 72, 84, 89

The cumulative frequency distribution table can be created as:

For plotting a less than type ogive the steps are given below:

• Mark the upper limit on the x-axis.
• Mark the cumulative frequency on the y-axis.
• Plot the points (x,y) using upper limits (x) and their corresponding cumulative frequency (y).
• Join the points using a freehand curve.