For a frequency distribution, the median and quartiles can be obtained by drawing the ogive of the distribution. Follow these steps.

**Step I:** Change the frequency distribution into a continuous
distribution by taking overlapping intervals. Let N be the total frequency.

**Step II:** Construct a cumulative-frequency table for the
distribution and draw the ogive accordingly by using proper scales of representation.

**Step III:** For median (i) If N is odd, find \(\frac{N + 1}{2}\),
and locate the point F on the y-axis which represents the cumulative frequency \(\frac{N
+ 1}{2}\).

(ii) If N is even, find the mean A of \(\frac{N}{2}\) and \(\frac{N}{2}\) + 1, which is given by A = \(\frac{1}{2}\){\(\frac{N}{2}\) + (\(\frac{N}{2}\) + 1)}. Locate the point F on the y-axis, which represents the cumulative frequency A.

For lower quartile: Find the integer c just greater than \(\frac{N}{4}\). Locate the point F on the y-axis, which represents the cumulative frequency c.

For upper quartile: Find the integer c just greater than \(\frac{3N}{4}\). Locate the point F on the y-axis, which represents the cumulative frequency c.

**Step IV:** Draw a line FD parallel to the x-axis to cut the
ogive at C.

**Step V:** Draw a line CM perpendicular to the x-axis
(class-interval axis) to cut the ogive at M. The variate represented by M is
the median or lower quartile or upper quartile as the case may be.

Solved Problems on Estimate Median, Quartiles from Ogive:

**1.** Estimate the median, lower quartile and upper quartile for
the following distribution.

**Class Interval**

0 - 10

10 - 20

20 - 30

30 - 40

40 - 50

50 - 60

**Frequency**

5

3

10

6

4

2

**Solution:**

Here, the distribution is continuous and total frequency = 30.

For constructing the ogive (step II), the following cumulative-frequency table is constructed.

**Class Interval**

0 - 10

10 - 20

20 - 30

30 - 40

40 - 50

50 - 60

**Frequency**

5

8

18

24

28

30

Take the following scales:

On the x-axis (class-interval axis), 1 cm = size 10.

On the y-axis (cumulative –frequency axis), 2 mm = frequency 1 (i.e., frequency of 1 is denoted by 2 mm).

Now, plot the pojnts (10, 5), (20, 8), (30, 18), (40, 24), (50, 28), (60, 30), and join them by a smooth curve to get the ogive.

Here, N = 30 = even. So, the mean of \(\frac{N}{2}\) and \(\frac{N}{2}\) + 1, i.e., the mean of 15 and 16, is 15.5. The point F on the y-axis represents the cumulative frequency 15.5. FC ∥ x-axis is drawn to cut the ogive at C. CM ⊥ x-axis is drawn to cut at M. The point M represents the median. Now, the point M represents the variate 28 on the x-axis.

So, the median is 28.

Now, \(\frac{N}{4}\) = \(\frac{30}{4}\) = 7.5. The
integer just greater than 7.5 is 8. The point F_{1} on the y-axis
represents the cumulative frequency 8. F_{1}C_{1} ∥
x-axis is drawn to cut the ogive at C_{1}. C_{1}Q_{1}
⊥
x-axis is drawn to cut the ogive at Q_{1}. The point Q_{1} represents
the lower quartile. Now, the point Q_{1} represents the variate 20. So,
the lower quartile is 20.

Next, \(\frac{3N}{4}\) = \(\frac{3 × 30}{4}\) = 22.5.
The integer just greater than 22.5 is 23. The point F_{2} on the
y-axis represents the cumulative frequency 23. F_{2}C_{2} ∥
x-axis is drawn to cut the ogive at C_{2}. C_{2}Q_{2}
⊥
x-axis is drawn to cut the ogive at Q_{2}. The point Q_{2} represents
the upperr quartile. Now, the point Q_{2} represents the variate 38.
So, the upper quartile is 38.

**Note:** Thse estimates are generally rough (that is, with
marginal error) because the drawing of an ogive is never perfect.

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