**Ogive (Cumulative frequency curve)**

** **

**Ogive** is a graphical representation
of the cumulative frequency distribution of continuous series.

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In a continuous series, if the upper
limit (or the lower limit) of each class interval is taken as x-coordinate and
its corresponding cumulative frequency (c.f.) as y-coordinate and the points
are plotted in the graph, we obtain a curve by joining the points with
freehand. Such a curve is known as the **ogive**
or **cumulative frequency curve**. It is
also called a **free hand curve**.

Since less than c.f. and more than c.f.
are the two types of cumulative frequencies, there are two types of ogives as
well. They are:

**1. ****Less than ogive (or less
than cumulative frequency curve)**

**2. ****More than ogive (or more
than cumulative frequency curve)**

**Less than ogive**

When the upper limit of each class interval is taken as
x-coordinate and its corresponding frequency as y-coordinate, the ogive so
obtained is known as less than ogive (or less than cumulative frequency curve).

Obviously, less than ogive is an increasing curve, sloping
upwards from left to right and has the shape of an elongated S.

**Construction of less than ogive:**

To construct a less than ogive, we
proceed through the following steps:

**1. **Make a less than cumulative frequency table.

**2. **Choose the suitable scale and make the upper class limit of each
class interval along the x-axis and cumulative frequencies along the y-axis.

**3. **Plot the coordinates (upper limit, less than c.f.) on the graph.

**4. **Join the points by freehand and obtain a less than ogive.

**More than ogive**

When the lower limit of each class interval is taken as x-coordinate
and its corresponding frequency as y-coordinate, the ogive so obtained is known
as more than ogive (or more than cumulative frequency curve).

More than ogive is a decreasing curve sloping downward from left
to right and has the shape of an elongated S, upside down.

**Construction of less than ogive:**

To construct a more than ogive, we
proceed through the following steps:

**1. **Make a more than cumulative frequency table.

**2. **Choose the suitable scale and make the lower class limit of each
class interval along the x-axis and cumulative frequencies along the y-axis.

**3. **Plot the coordinates (lower limit, more than c.f.) on the graph.

**4. **Join the points by freehand and obtain a more than ogive.

**Examples:**

**Example 1: **The table given below shows the marks
obtained by 80 students in science. Construct (i) less than ogive (ii) more than ogive.

**Solution:**
Here,

Less than
cumulative frequency distribution table:

Here, we have the coordinates to draw less than ogive: (10, 3), (20,
11), (30, 28), (40, 57), (50, 72), (60, 78) and (70, 80).

Plotting these points on a graph, we have
the following less than ogive.

More than cumulative frequency distribution table:

Here, we have the coordinates to draw
more than ogive: (0, 80), (10, 77), (20, 69), (30, 52), (40, 23), (50, 8) and
(60, 2).

Plotting these points on a graph, we have
the following more than ogive.

**Less than and more than (Combined) Ogive**

If we draw both less than and more than
ogive of a distribution on the same graph paper, it is called **combined ogive**. The
point of intersection of two ogives in a combined ogive is called the
**vital-point**.

The foot of the perpendicular drawn from vital-point to x-axis
gives the value of the median of the distribution. And, the corresponding class is
the proper class for the median.

**Uses of ogive**

Following are the uses of cumulative
frequency curve (ogive):

**1. **Ogive or cumulative frequency curve helps us to determine as well
as portray the number or proportion of cases above or below the given value.

**2. **Median (Q_{2}) and quartiles i.e. lower quartile (Q_{1})
and upper quartile (Q_{3}) can be estimated from the ogive of a
frequency distribution.

**Examples:**

**Example 2:** Construct the combined ogive from the
data given in the table of Example 1,
and find the median.

**Solution:**
Here,

Less than and
more than cumulative frequency distribution table:

Here, we have the coordinates to draw
less than ogive: (10, 3), (20, 11), (30, 28), (40, 57), (50, 72), (60, 78) and
(70, 80).

Here, we have the coordinates to draw
more than ogive: (0, 80), (10, 77), (20, 69), (30, 52), (40, 23), (50, 8) and
(60, 2).

Plotting these points on a graph, we have
the following less than and more than combined ogive.

A perpendicular is drawn from the point of intersection of two ogives
which meets the x-axis at 34.14 (approx.) units from the origin. So, the required
median of the given distribution is 34.14.

**Example 3:** The table given below
shows the marks obtained by 60 students in mathematics. Construct a less than
ogive and compute median, first quartile (Q_{1}) and third quartile (Q_{3}).

**Solution:** Here,

Less than cumulative frequency distribution table,

Here, we have the coordinates to draw less than ogive: (10, 4), (20, 14),
(30, 34), (40, 49), (50, 55) and (60, 60).

Plotting these points on a graph, we have
the following less than ogive.

Median lies in 50% of the total data.

∴ 50% of 60 students = 30 students

Q_{1}
lies in 25% of the total data.

∴ 25% of 60 students = 15 students

Q_{3}
lies in 75% of the total data.

∴ 75% of 60 students = 45 students

Straight lines
drawn from 30, 15, and 45 (frequency) on y-axis parallel to x-axis intersects
the curve at A, B, and C. And, perpendiculars drawn from A, B, and C meets the x-axis
at P, Q, and R with x-coordinates 28, 20.5, and 37.33 respectively.

Therefore,

Median = 28
(approx.)

Q_{1}
= 20.5 (approx.)

Q_{3}
= 37.33 (approx.)

**Example 4:** Draw the less than
ogive from the following data and the questions.

a. Find the no. of workers having wages more than 40.

b. Find the no. of
workers having wages less than 80.

c. Find the no. of
workers having wages between 80 and 100.

**Solution:** Here,

Less than cumulative frequency distribution table,

Here, we have the coordinates to draw less than ogive: (20, 5), (40,
13), (60, 23), (80, 32) and (100, 35).

Plotting these points on a graph, we have
the following less than ogive.

From the graph,

a. No. of workers
having wages more than 40 = (35 – 13) = 22

b. No. of workers
having wages less than 80 = 32

c. No. of workers
having wages between 80 and 100 = 3

**Example 5:** From the given
cumulative frequency curve (ogive), determine the lower quartile (Q_{1})
class, median (Q_{2}) class and upper quartile (Q_{3}) class.

**Solution:** From the given
graph,

Total number of data (N) = 40

Passing
the lines from 10, 20, and 30 of the y-axis (i.e cumulative frequencies) parallel to
x-axis, we get the corresponding points on the ogive curve. And, again vertical
lines from those points give the required class for Q_{1}, Q_{2}
and Q_{3}.

∴ Q_{1} class
= 10-20

∴ Q_{2} class
= 30-40

∴ Q_{3} class
= 40-50

*Do you have any questions regarding the ogive?*

*Do you have any questions regarding the ogive?*

* *

*You can ask your
questions or problems here, in the comment section below.*

For a continuous data distribution ,10-20 with frequency 10,20-30 with frequency 24 & 30-40 with frequency 43, at which point more than cumulative ogive will start.

ReplyDeleteIt starts from the point (10, 77). Here is the solution to your problem,

Delete[image src="https://1.bp.blogspot.com/-djMsNwDGlII/YBwZPi-nIzI/AAAAAAAAHk4/nSfZY1o8vioLVmzsUGyua9ieQfg9kFzBgCLcBGAsYHQ/s16000/comment%2Bsolution%2B-%2Bogive.png"/]

If you have more problems regarding the ogive (cumulative frequency curve), comment here.