Ogives: Less Than & More Than Types

Ogives, also known as cumulative frequency graphs, are graphical representations of cumulative frequency distributions. They are used to visually represent the cumulative frequencies of a dataset. There are two main types of ogives:

• Less than type ogive: This type shows the cumulative frequency of data points that are less than or equal to a particular value.
• More than type ogive: This type shows the cumulative frequency of data points that are greater than or equal to a particular value.

Both types of ogives are useful for determining measures like the median, quartiles, and percentiles, as well as for comparing different datasets.

Formulae

There are no specific formulas to *calculate* an ogive. The process involves constructing a cumulative frequency table and then plotting the cumulative frequencies against the corresponding class boundaries (for Less than type) or lower class limits (for More than type).

• Less than type: The cumulative frequency for each class is the sum of the frequencies of all classes up to and including that class.
• More than type: The cumulative frequency for each class is the sum of the frequencies of all classes from that class up to the last class.

Examples

Example-1: Less than type ogive

Consider the following data representing the marks obtained by students in a test:

Marks	Number of Students (Frequency)
0-10	5
10-20	8
20-30	12
30-40	10
40-50	5

To create a “Less than type” ogive, we first calculate the cumulative frequency:

Marks	Number of Students (Frequency)	Cumulative Frequency (Less than)
Less than 10	5	5
Less than 20	8	13 (5+8)
Less than 30	12	25 (13+12)
Less than 40	10	35 (25+10)
Less than 50	5	40 (35+5)

We then plot the cumulative frequencies against the upper class boundaries (10, 20, 30, 40, and 50). The resulting graph is the “Less than type” ogive.

Example-2: More than type ogive

Using the same data as Example-1:

Marks	Number of Students (Frequency)
0-10	5
10-20	8
20-30	12
30-40	10
40-50	5

To create a “More than type” ogive, we calculate the cumulative frequency (More than):

Marks	Number of Students (Frequency)	Cumulative Frequency (More than)
More than 0	5	40 (5+8+12+10+5)
More than 10	8	35 (8+12+10+5)
More than 20	12	27 (12+10+5)
More than 30	10	15 (10+5)
More than 40	5	5

We then plot the cumulative frequencies against the lower class boundaries (0, 10, 20, 30, and 40). The resulting graph is the “More than type” ogive.

Common mistakes by students

• Confusing class boundaries with class limits: Students sometimes plot the cumulative frequencies against the wrong values (either class limits or upper class boundary instead of lower class boundary for more than type).
• Incorrectly calculating cumulative frequencies: A common error is miscalculating the cumulative frequencies. For “Less than” type, they may add frequencies incorrectly, or for “More than” type, they may start from the beginning.
• Not labeling the axes properly: Failing to label the axes of the ogive graphs correctly, including appropriate units and titles, can lead to misinterpretations.
• Interpreting the ogive incorrectly: Students might struggle to correctly read values from the ogive, such as the median or quartiles.

Real Life Application

Ogives are used in various real-life applications, including:

• Quality Control: Analyzing the distribution of product measurements to identify defects.
• Financial Analysis: Understanding the distribution of incomes or investment returns.
• Demographics: Studying the distribution of ages in a population.
• Education: Analyzing the distribution of test scores to evaluate student performance or to set grading scales.
• Healthcare: Analyzing patient’s recovery data.

Fun Fact

The ogive is a valuable tool because it allows us to quickly visualize the distribution of data and estimate key statistical measures like the median and quartiles without needing to calculate them precisely from the raw data. The intersection point of “less than” and “more than” ogives gives us the median of the dataset.

Recommended YouTube Videos for Deeper Understanding

Q.1 The following data represents the marks obtained by 40 students in a test: Marks: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60 Number of Students: 4, 6, 7, 10, 8, 5 What is the median mark using the less than ogive?

Check Solution

Ans: C

Calculate the cumulative frequencies. Then, locate the median by finding the value corresponding to the point where the cumulative frequency is 20 (40/2) on the ogive.

Q.2 The following data represents the distribution of weights (in kg) of 50 students: Weight: 40-45, 45-50, 50-55, 55-60, 60-65 Number of Students: 4, 10, 16, 12, 8 What is the lower quartile ($Q_1$) from the less than ogive?

Check Solution

Ans: A

Find the cumulative frequencies. Locate the lower quartile by finding the value corresponding to the point where the cumulative frequency is 12.5 (50/4) on the ogive.

Q.3 Given the following data: Age (in years): 10-20, 20-30, 30-40, 40-50 Number of people: 5, 10, 15, 20 If we plot a more than ogive, which point will be plotted for the first class interval?

Check Solution

Ans: B

For a more than ogive, we plot the lower limits of the class intervals against the cumulative frequencies from the beginning. The first point is (lower limit of the first class, total frequency)

Q.4 The following data represents the daily wages (in Rs.) of workers: Wages: 100-120, 120-140, 140-160, 160-180, 180-200 Number of Workers: 5, 10, 15, 8, 2 What is the upper quartile ($Q_3$) from the more than ogive?

Check Solution

Ans: B

Find the cumulative frequencies starting from the end. Locate the upper quartile by finding the value corresponding to the point where the cumulative frequency is 12.5 (50-37.5) on the more than ogive.

Q.5 The following data represents the heights of students: Height (cm): 140-150, 150-160, 160-170, 170-180 Number of students: 8, 12, 10, 6 What is the range of heights for the interval within which the median lies when plotted using a ‘less than’ ogive?

Check Solution

Ans: B

Calculate the cumulative frequencies. The median lies in the interval where the cumulative frequency crosses the middle value, which is 18 (36/2).

Next Topic: Empirical Relationship: Mean, Median, Mode