Ch 13: Statistics - Median & Ogives
Master the Median formula and Graphical Representation of Cumulative Frequency (Ogives) from Old Ex 14.4.
Exercise 13.3 Solutions
💡 Formula: Median = $l + \left[ \frac{(N/2) - cf}{f} \right] \times h$
Q1. Monthly consumption of electricity of 68 consumers. Find Median, Mean, and Mode.
Solution:
Total frequency $N = 68 \implies N/2 = 34$.
Cumulative frequency just greater than 34 is 42, which belongs to class 125-145.
Median class = 125-145. $l = 125$, $cf = 22$, $f = 20$, $h = 20$.
Median = $125 + [ (34 - 22) / 20 ] \times 20 = 125 + 12 =$ 137 units.
Mode = $125 + [ (20 - 13) / (40 - 13 - 14) ] \times 20 = 125 + 10.76 =$ 135.76 units.
Mean (using empirical formula: 3 Median = Mode + 2 Mean) $\implies$ Mean = 137.05 units.
Q2. If the median of the distribution is 28.5, find the values of x and y. Total frequency is 60.
Solution:
Sum of frequencies $N = 60$. From table: $45 + x + y = 60 \implies$ $x + y = 15$ --- (Eq 1)
Median is 28.5, which lies in class 20-30. Median class = 20-30.
$l = 20, N/2 = 30, cf = 5 + x, f = 20, h = 10$.
Median = $l + [ (N/2 - cf) / f ] \times h$
$28.5 = 20 + [ (30 - (5 + x)) / 20 ] \times 10$
$8.5 = (25 - x) / 2$
$17 = 25 - x \implies$ $x = 8$.
Put $x = 8$ in Eq 1: $8 + y = 15 \implies$ $y = 7$.
Q3. Life insurance agent age distribution (Below 20, Below 25...). Find median age.
Solution:
This is a "less than" cumulative frequency table. First, convert it to normal classes:
15-20, 20-25, 25-30...
Frequencies: 2, (6-2)=4, (24-6)=18, (78-24)=54...
$N = 100 \implies N/2 = 50$. Median class is 35-40.
$l = 35, cf = 45, f = 33, h = 5$.
Median = $35 + [ (50 - 45) / 33 ] \times 5 = 35 + (25 / 33) = 35 + 0.76 =$ 35.76 years.
Q4. Lengths of 40 leaves. Find median length. (118-126, 127-135...)
Note: Make classes continuous (117.5-126.5).
$N/2 = 20$. Median class = 144.5 - 153.5.
$l = 144.5, cf = 17, f = 12, h = 9$.
Median = $144.5 + [ (20 - 17) / 12 ] \times 9 =$ 146.75 mm.
Q5. Lifetimes of 400 neon lamps. Find median.
$N/2 = 200$. Median class = 3000-3500.
$l = 3000, cf = 130, f = 86, h = 500$.
Median = $3000 + [ (200 - 130) / 86 ] \times 500 =$ 3406.98 hours.
Q6. 100 surnames letters. Find Median, Mean, Mode.
$N/2 = 50$. Median class = 7-10.
Median = $7 + [ (50 - 36) / 40 ] \times 3 =$ 8.05 letters.
Mode = $7 + [ (40 - 30) / (80 - 30 - 16) ] \times 3 =$ 7.88 letters.
Mean = 8.32 letters.
Q7. Weights of 30 students. Find median weight.
$N/2 = 15$. Median class = 55-60.
$l = 55, cf = 13, f = 6, h = 5$.
Median = $55 + [ (15 - 13) / 6 ] \times 5 = 55 + 1.67 =$ 56.67 kg.
Old Ex 14.4 (Ogives / Graphical Method)
💡 Trick: Less than Ogive uses Upper Class Limits. More than Ogive uses Lower Class Limits. Median is the x-coordinate where the two ogives intersect.
Q1. Convert the distribution to a "less than type" cumulative frequency distribution, and draw its ogive.
Solution:
Upper Limits: 120, 140, 160, 180, 200.
Cumulative Frequencies (cf): 12, 26, 34, 40, 50.
To draw Ogive: Plot points (120, 12), (140, 26), (160, 34), (180, 40), (200, 50) on a graph and join them with a smooth freehand curve.
Q2. Draw a less than type ogive for the given data. Hence obtain the median weight from the graph and verify the result by using the formula.
Solution:
Points to plot: (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), (50, 32), (52, 35).
Graph Median: Total N = 35, so N/2 = 17.5. Locate 17.5 on the y-axis, draw a horizontal line to the curve, and drop a perpendicular to the x-axis. It hits at x = 46.5 kg.
Formula Verification: Median class is 46-48. $l=46, f=14, cf=14, h=2$.
Median = $46 + [ (17.5 - 14) / 14 ] \times 2 = 46 + 0.5 =$ 46.5 kg. Verified.
Q3. Change the distribution to a "more than type" distribution, and draw its ogive.
Solution:
For more than type, use lower limits and subtract frequencies from the total.
- More than or equal to 50: 100
- More than or equal to 55: 100 - 2 = 98
- More than or equal to 60: 98 - 8 = 90
- More than or equal to 65: 90 - 12 = 78
- More than or equal to 70: 78 - 24 = 54
- More than or equal to 75: 54 - 38 = 16
To draw Ogive: Plot (50, 100), (55, 98), (60, 90), (65, 78), (70, 54), (75, 16) and join with a smooth downward curve.