Estimate Median, Quartiles from Ogive

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.

Solution:

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

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

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 F₁ on the y-axis represents the cumulative frequency 8. F₁C₁ ∥ x-axis is drawn to cut the ogive at C₁. C₁Q₁ ⊥ x-axis is drawn to cut the ogive at Q₁. The point Q₁ represents the lower quartile. Now, the point Q₁ 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 F₂ on the y-axis represents the cumulative frequency 23. F₂C₂ ∥ x-axis is drawn to cut the ogive at C₂. C₂Q₂ ⊥ x-axis is drawn to cut the ogive at Q₂. The point Q₂ represents the upperr quartile. Now, the point Q₂ 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.

9th Grade Math

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