What is an Ogive (Cumulative Frequency Curve)? – Meaning, Types & Uses

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The word ‘Ogive’ is a term used in architecture meaning a pointed or Gothic arch. In statistics, the word ‘Ogive’ is used for a cumulative frequency graph. Ogive graphs are used to find the median of a frequency distribution. There are two types of ogive graphs in statistics – more than ogive and less than ogive.

Let’s understand what is an ogive (or cumulative frequency) curve and how it is drawn and used to find the median of the frequency distribution.

What is an Ogive in Statistics?

An ogive, sometimes called a cumulative frequency polygon, is a type of frequency polygon that shows cumulative frequencies.  An ogive graph plots cumulative frequency on the $y$-axis and class boundaries along the $x$-axis. It’s very similar to a histogram, only instead of rectangles, an ogive has a single point marking where the top right of the rectangle would be. It is usually easier to create this kind of graph from a frequency table.

Mostly ogive is used to illustrate the data in the pictorial representation. It helps in estimating the number of observations that are less than or equal to the particular value. They are even used to calculate the median, percentiles, and the summary of five-number data. Here, the median is a basic value that gives us data about the middle of the data given. Thus, considering ogive we get information about the values of data that are above or below a certain allotted median making it feasible to find the median. 

Based on the way an ogive is plotted it can be either less than ogive or more than ogive.

Before moving on to ogive, let’s first understand the central concept – cumulative frequency.

What is a Cumulative Frequency Distribution?

A cumulative frequency distribution is the sum of the class interval and all class intervals below it in a frequency distribution. All that means is you’re adding up a value and all of the values that came before it.

There are two types of cumulative frequency distributions.

• Less Than Cumulative Distribution
• More Than Cumulative Distribution

The first step is to check whether the frequency distribution has an exclusive class interval or an inclusive class interval.

Inclusive Class Interval

The inclusive class interval is used for discrete data. In an inclusive class interval, the lower limit of a class does not get repeated in the upper limit of the preceding class. The interval $\text{a} – \text{b}$ in inclusive class interval includes all values between $a$ and $b$ including both $a$ and $b$. In inequality form, it can also be represented as $a \le x \le b$.

For example, the inclusive class interval $5 – 10$ includes all the values from $5$ to $10$ including $5$ and $10$, i.e., the values included in the inclusive class interval $5 – 10$ include values $5$, $6$, $7$, $8$, $9$ and $10$. 

Note: Inclusive class interval includes only finite fixed values.

Exclusive Class Interval

The exclusive class interval is used for continuous data. In an exclusive class interval, the lower limit of a class is repeated in the upper limit of the preceding class. The interval $\text{a} – \text{b}$ in exclusive class interval includes all values between $a$ and $b$ including both $a$ but excluding $b$. In inequality form, it can also be represented as $a \le x \lt b$.

For example, the exclusive class interval $5 – 10$ includes all the values from $5$ to $10$ including $5$ but excluding $10$, i.e., the values included in the inclusive class interval $5 – 10$ include values $5$, $5.4$, $6$, $6.89$, $7$, $8$, $9$, $9.987$, $9.999$, and so on. 

Note: Exclusive class interval includes infinite values within the class interval.

Converting Inclusive Class Interval to Exclusive Class Interval

The steps followed to convert an inclusive class interval to an exclusive class interval are:

Step 1: Find the difference between the upper limit of a class interval and the lower limit of the next class interval.

Step 2: Divide the difference obtained in Step 1 by 2.

Step 3: Add the number obtained in Step 2 to the upper limit of each class interval.

Step 4: Subtract the number obtained in Step 2 from the lower limit of each class interval.

Step 5: The class intervals obtained are exclusive class intervals.

Examples

Ex 1: The following frequency distribution is in inclusive class interval form. Convert it to exclusive class interval form.

Select any class interval (Other than the first or last).

Selected class interval $30 – 39$.

Upper limit of $30 – 39$ is $39$.

Class interval next to $30 – 39$ is $40 – 49$.

Lower limit of $40 – 49$ is $40$.

Difference of $39$ and $40$ is $40 – 39 = 1$.

Dividing $1$ by $2$, we get $\frac{1}{2} = 0.5$.

Now, add $0.5$ to an upper limit of all class intervals and subtract $0.5$ from a lower limit of all class intervals.

$10 – 19$ becomes $10 – 0.5 – 19 + 0.5 = 9.5 – 19.5$

$20 – 29$ becomes $20 – 0.5 – 29 + 0.5 = 19.5 – 29.5$

$30 – 39$ becomes $30 – 0.5 – 39 + 0.5 = 29.5 – 39.5$

$40 – 49$ becomes $40 – 0.5 – 49 + 0.5 = 39.5 – 49.5$

$50 – 59$ becomes $50 – 0.5 – 59 + 0.5 = 49.5 – 59.5$

After setting the frequency distribution table in exclusive class interval form, find the cumulative frequency of each of the class intervals.

Less Than Cumulative Distribution

Less than cumulative frequency is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The process of summing 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.

The steps followed to compute less than cumulative frequencies.

Step 1: Add one more column to the right of the Frequency column for Less Than Cumulative Frequency.

Step 2: Copy the frequency to the less than cumulative frequency column of the first interval.

Step 3: Move to the next class interval.

Step 4: Add the frequency of the current class interval to the less than cumulative frequency of the preceding class interval.

Step 5: The sum obtained in Step 4 is the less than cumulative frequency of the current class interval.

Step 6: Repeat Step 3 to Step 5, till the last class interval is reached.

Step 7: The table obtained is the Less Than Cumulative Frequency table.

Note: The less than cumulative frequency of the last class interval is always equal to the sum of all frequencies.

More Than Cumulative Distribution

More 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 more than cumulative frequency.

The steps followed to compute more than cumulative frequencies.

Step 1: Add one more column to the right of the Frequency column for Greater Than Cumulative Frequency.

Step 2: Copy the frequency of the last interval to the more than cumulative frequency column of the first interval.

Step 3: Move to the next class interval.

Step 4: Subtract the frequency of the current class interval from the more than cumulative frequency of the preceding class interval.

Step 5: The difference obtained in Step 4 is the more than cumulative frequency of the current class interval.

Step 6: Repeat Step 3 to Step 5, till the last class interval is reached.

Step 7: The table obtained is the More Than Cumulative Frequency table.

Note: The more than cumulative frequency of the last class interval is always equal to the frequency of the first class interval.

What is Ogive Curve in Statistics?

The graphs of the distribution are frequency graphs that exhibit the characteristics of discrete and continuous data. Such figures are more appealing to the attention than the tabulated data. It helps to facilitate the comparison study for two or more frequency distributions. We can relate the form and pattern of the 2 frequency distributions. The two methods of Ogives are

• Less Than Ogive
• More Than or Greater Than Ogive

In the above figure, the curve on the left (cyan colour) is the ‘Less Than Ogive’ and the curve on the right (orange colour) is the ‘More Than or Greater Than Ogive’.

What is Less Than Ogive Curve?

In the Less Than Ogive or Less Than Cumulative Frequency Curve, 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 limits of the class intervals on the $x$-axis

Step 2: Mark the cumulative frequencies on the $y$-axis

Step 3: Plot the points $\left(x, y \right)$ using upper limits $\left(x \right)$ and their corresponding Cumulative frequency $\left(y \right)$

Step 4: Join the points by a smooth freehand curve

What is More Than Ogive Curve?

In the More Than Ogive or More Than Cumulative Frequency Curve, 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 the 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 limits of the class intervals  on the $x$-axis

Step 2: Mark the cumulative frequencies on the $y$-axis

Step 3: Plot the points $\left(x, y \right)$ using lower limits $\left(x \right)$ and their corresponding Cumulative frequency $\left(y \right)$

Step 4: Join the points by a smooth freehand curve

Finding Median Graphically

We can find the median of a frequency distribution by using any of the two types of ogives or using both to find the median of a frequency distribution.

Finding Median Using Less Than Ogive

A less than ogive is used to find the median of a frequency distribution. The steps to compute the median of a frequency distribution using a less than ogive are:

Step 1: Check whether the class intervals in the frequency distribution are of an inclusive type or exclusive type

Step 2: If the class intervals are of inclusive type, convert them to exclusive class intervals

Step 3: Convert frequency distribution to less than cumulative frequency distribution

Step 4: Plot the points $\left(x, y \right)$, $x$ representing more than class interval and $y$ representing respective cumulative frequencies

Step 5: Join the points by a free-hand curve to get a less than ogive

Step 6: Compute $\frac{n}{2}$, where $n$ is the sum of all frequencies

Step 7: Mark a point $\text{M}$, whose ordinate is $\frac{n}{2}$ and lies on the curve

Step 8: Note down the abscissa of the point $\text{M}$

Step 9: The abscissa ($x$-coordinate) of the point $\text{M}$ is the median of the frequency distribution

Examples

Ex 1: Find the median of the given distribution graphically using Less Than Ogive.

Convert the given frequency distribution to less than frequency distribution.

Plot the points $\left(x, y \right)$, $x$ from the less than class intervals, and $y$ from the less cumulative frequencies. Join the points by a free-hand curve.

Sum of frequencies $n = 29$.

Therefore, $\frac{n}{2} = \frac{29}{2} = 14.5$.

Locate a point on the ogive whose ordinate ($y$-coordinate) is $14.5$.

Median = $33.25$

Finding Median Using More Than Ogive

A more than ogive is used to find the median of a frequency distribution. The steps to compute the median of a frequency distribution using a more than ogive are:

Step 1: Check whether the class intervals in the frequency distribution are of an inclusive type or exclusive type

Step 2: If the class intervals are of inclusive type, convert them to exclusive class intervals

Step 3: Convert frequency distribution to more than cumulative frequency distribution

Step 4: Plot the points $\left(x, y \right)$, $x$ representing less than class interval and $y$ representing respective cumulative frequencies

Step 5: Join the points by a free-hand curve to get a more than ogive

Step 6: Compute $\frac{n}{2}$, where $n$ is the sum of all frequencies

Step 7: Mark a point $\text{M}$, whose ordinate is $\frac{n}{2}$ and lies on the curve

Step 8: Note down the abscissa of the point $\text{M}$

Step 9: The abscissa ($x$-coordinate) of the point $\text{M}$ is the median of the frequency distribution

Examples

Ex 1: Find the median of the given distribution graphically using Less Than Ogive.

Convert the given frequency distribution to more than frequency distribution.

Plot the points $\left(x, y \right)$, $x$ from the more than class intervals, and $y$ from the more than cumulative frequencies. Join the points by a free-hand curve.

Sum of frequencies $n = 29$.

Therefore, $\frac{n}{2} = \frac{29}{2} = 14.5$.

Locate a point on the ogive whose ordinate ($y$-coordinate) is $14.5$.

Median = $33.25$.

Finding Median Using Both Less Than and More Than Ogives

The steps to compute the median of a frequency distribution using both less than and more than ogives are:

Step 1: Draw a less than ogive

Step 2: Draw a more than ogive

Step 3: Locate the point of intersection of the two ogives

Step 4: The abscissa ($x$-coordinate) of the point is the median of the frequency distribution

Practice Problems

1. The following distribution shows the marks obtained by students in a class. Draw a less than ogive for the distribution and hence find the median marks.

2. The following distribution shows the marks obtained by students in a class. Draw a more than ogive for the distribution and hence find the median marks.

3. The following distribution shows the marks obtained by students in a class. Draw a less than ogive and a more than ogive for the distribution and hence find the median marks.

FAQs

Conclusion

An ogive is a type of frequency polygon that shows cumulative frequencies. There are two types of ogives – less than ogive and more than ogive. The ogives are used to find the median of a frequency distribution graphically.

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