Exercise - 14.3
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.
| Monthly consuption (in units) | Number of consumers | Cumulative Frequency (cf) |
| 65 - 85 | 4 | 4 |
| 85 - 105 | 5 | 4 + 5 = 9 |
| 105 - 125 | 13 | 9 + 13 = 22 |
| 125 - 145 | 20 | 22 + 20 = 42 |
| 145 - 165 | 14 | 42 + 14 = 56 |
| 165 - 185 | 8 | 56 + 8 = 64 |
| 185 - 205 | 4 | 64 + 4 = 68 |
| Total | n = 68 | |
Now, n = 68
n/2 = 68/2 = 34 is 42. So, this observation lies in the class 125 - 145.
Then, l (the lower limit) = 125,
cf (the cumulative frequency of the class preceding 125 - 145) = 22,
f (the frequency of the median class 125 - 145) = 20,
h (the class size) = 20.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 125 + [(68/2 - 22)/20] x 20
= 125 + [(34 - 22)/20] x 20
= 125 + (12/20 x 20)
= 125 + 12
= 137
So, the median of data is 137.
From given table, to find the class mark (xi) for each inerval, the following relation is used.
Class mark (xi) = (upper class limit + lower class limit)/2
Class size (h) = 20
Taking assumed mean (a) = 135
| Monthly consuption (in units) | Number of consumers (fi) | xi | di = xi - 135 | ui = di/20 | fiui |
| 65 - 85 | 4 | 75 | -60 | -3 | -12 |
| 85 - 105 | 5 | 95 | -40 | -2 | -10 |
| 105 - 125 | 13 | 115 | -20 | -1 | -13 |
| 125 - 145 | 20 | 135 | 0 | 0 | 0 |
| 145 - 165 | 14 | 155 | 20 | 1 | 14 |
| 165 - 185 | 8 | 175 | 40 | 2 | 16 |
| 185 - 205 | 4 | 195 | 60 | 3 | 12 |
| Total | Σfi = 68 | | | | Σfiui = 7 |
Now,
x = a + (Σf
iu
i/Σf
i) x h = 135 + (7/68) x 20
= 135 + 140/68
= 135 + 2.058
= 137.058
Therefore, the mean of data is 137.058.
Here the maximum class frequency is 20, and the class corresponding to this frequency is 125 – 145.
So, the modal class is 125 – 145.
Now, Modal class = 125 – 145,
lower limit(l) of modal class = 125,
class size(h) = 20
frequency(f1) of modal class = 20,
frequency(f0) of modal class = 13,
frequency(f2) of modal class = 14,
Therefore, Mode = l + [(f1 - f0)/(2f1 - f0 - f2)] x h
= 125 + [(20-13)/{(2x20)-13-14}] x 20
= 125 + [7/(40-27)] x 20
= 125 + (7/13 x 20)
= 125 + 140/13
= 125 + 10.76
= 135.76
Therefore, the mode of this data is 135.76.
Hence, median, mean and mode of the given data is 137, 137.05 and 135.76 respectively.
The three measures are approximately the same in this case.
If the median of the distribution given below is 28.5, find the values of x and y.
| Class interval | Frequency | Cumulative Frequency (cf) |
| 0 - 10 | 5 | 5 |
| 10 - 20 | x | 5 + x |
| 20 - 30 | 20 | 25 + x |
| 30 - 40 | 15 | 40 + x |
| 40 - 50 | y | 40 + x + y |
| 50 - 60 | 5 | 45 + x + y |
| Total | n = 45 + x + y | |
Given that: n = 60
Now, n = 45 + x + y
60 = 45 + x + y
x + y = 60 - 45
x + y = 15 ........(i)
The median is 28.5, which lies in the class 20 – 30
So, l = 20, f = 20, cf = 5 + x, h = 10
Therefore, Median = l + [(n/2 - cf)/f] x h
28.5 = 20 + [{(60/2) - (5 + x)}/20] x 10
28.5 = 20 + [(30 - 5 - x)/20] x 10
28.5 - 20 = [(25 - x)/20] x 10
8.5 = (25 - x)/2
17 = 25 - x
x = 25 - 17
x = 8
Therefore, from (i), we get
8 + y = 15
y = 15 - 8
y = 7
Hence, the values of x and y are 8 and 7 respectively.
A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.
Here, class width is not the same.
There is no requirement of adjusting the frequencies according to class intervals.
The given frequency table is of less than type represented with upper class limits.
The policies were given only to the persons with age 18 years onwards but less than 60 years.
| Age (in years) | Number of policy holders | Cumulative Frequency (cf) |
| 18 - 20 | 2 | 2 |
| 20 - 25 | 6 - 2 = 4 | 2 + 4 = 6 |
| 25 - 30 | 24 - 6 = 18 | 6 + 18 = 24 |
| 30 - 35 | 45 - 24 = 21 | 24 + 21 = 45 |
| 35 - 40 | 78 - 45 = 33 | 45 + 33 = 78 |
| 40 - 45 | 89 - 78 = 11 | 78 + 11 = 89 |
| 45 - 50 | 92 - 89 = 3 | 89 + 3 = 92 |
| 50 - 55 | 98 - 92 = 6 | 92 + 6 = 98 |
| 55 - 60 | 100 - 98 = 2 | 98 + 2 = 100 |
| Total | n = 100 | |
Now, n = 100
n/2 = 100/2 = 50 is 78. So, this observation lies in the class 35 - 40.
Then, l (the lower limit) = 35,
cf (the cumulative frequency of the class preceding 35 - 40) = 45,
f (the frequency of the median class 35 - 40) = 33,
h (the class size) = 5.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 35 + [(100/2 - 45)/33] x 5
= 35 + [(50 - 45)/33] x 5
= 35 + (5/33 x 5)
= 35 + 25/33
= 35 + 0.76
= 35.76
So, the median of data is 35.76 years.
The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table:
Find the median length of the leaves.
(Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . ., 171.5 - 180.5.)
It can be observed that class intervals are not continuous. There is a gap of 1 between two class intervals.
Therefore, 1/2 has to be added to upper class limit and 1/2 has to be substracted from the lower class limit of each interval.
| Length (in mm) | Number of leaves | Cumulative Frequency (cf) |
| 117.5 - 126.5 | 3 | 3 |
| 126.5 - 135.5 | 5 | 3 + 5 = 8 |
| 135.5 - 144.5 | 9 | 8 + 9 = 17 |
| 144.5 - 153.5 | 12 | 17 + 12 = 29 |
| 153.5 - 162.5 | 5 | 29 + 5 = 34 |
| 162.5 - 171.5 | 4 | 34 + 4 = 38 |
| 171.5 - 180.5 | 2 | 38 + 2 = 40 |
| Total | n = 40 | |
Now, n = 40
n/2 = 40/2 = 20 is 29. So, this observation lies in the class 144.5 - 153.5.
Then, l (the lower limit) = 144.5,
cf (the cumulative frequency of the class preceding 144.5 - 153.5) = 17,
f (the frequency of the median class 144.5 - 153.5) = 12,
h (the class size) = 9.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 144.5 + [(40/2 - 17)/12] x 9
= 144.5 + [(20 - 17)/12] x 9
= 144.5 + (3/12 x 9)
= 144.5 + 9/4
= 144.5 + 2.25
= 146.75
Therefore, the median length of leaves is 146.75 mm.
The following table gives the distribution of the life time of 400 neon lamps :
Find the median life time of a lamp.
| Life time (in hours) | Number of lamps | Cumulative Frequency (cf) |
| 1500 - 2000 | 14 | 14 |
| 2000 - 2500 | 56 | 14 + 56 = 70 |
| 2500 - 3000 | 60 | 70 + 60 = 130 |
| 3000 - 3500 | 86 | 130 + 86 = 216 |
| 3500 - 4000 | 74 | 216 + 74 = 290 |
| 4000 - 4500 | 62 | 290 + 62 = 352 |
| 4500 - 5000 | 48 | 352 + 48 = 400 |
| Total | n = 400 | |
Now, n = 400
n/2 = 400/2 = 200 is 216. So, this observation lies in the class 3000 - 3500.
Then, l (the lower limit) = 3000,
cf (the cumulative frequency of the class preceding 3000 - 3500) = 130,
f (the frequency of the median class 3000 - 3500) = 86,
h (the class size) = 500.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 3000 + [(400/2 - 130)/86] x 500
= 3000 + [(200 - 130)/86] x 500
= 3000 + (70/86 x 500)
= 3000 + 3500/86
= 3000 + 406.976
= 3406.976
So, the median life time of lamps is 3406.976 hours.
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:
Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.
| Number of letters | Number of surnames | Cumulative Frequency (cf) |
| 1 - 4 | 6 | 6 |
| 4 - 7 | 30 | 6 + 30 = 36 |
| 7 - 10 | 40 | 36 + 40 = 76 |
| 10 - 13 | 16 | 76 + 16 = 92 |
| 13 - 16 | 4 | 92 + 4 = 96 |
| 16 - 19 | 4 | 96 + 4 = 100 |
| Total | n = 100 | |
Now, n = 100
n/2 = 100/2 = 50 is 76. So, this observation lies in the class 7 - 10.
Then, l (the lower limit) = 7,
cf (the cumulative frequency of the class preceding 7 - 10) = 36,
f (the frequency of the median class 7 - 10) = 40,
h (the class size) = 3.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 7 + [(100/2 - 36)/40] x 3
= 7 + [(50 - 36)/40] x 3
= 7 + (14/40 x 3)
= 7 + 42/40
= 7 + 1.05
= 8.05
So, the median of data is 8.05.
From given table, to find the class mark (xi) for each inerval, the following relation is used.
Class mark (xi) = (upper class limit + lower class limit)/2
Class size (h) = 3
Taking assumed mean (a) = 11.5
| Number of letters | Number of surnames (fi) | xi | di = xi - 11.5 | ui = di/3 | fiui |
| 1 - 4 | 6 | 2.5 | -9 | -3 | -18 |
| 4 - 7 | 30 | 5.5 | -6 | -2 | -60 |
| 7 - 10 | 40 | 8.5 | -3 | -1 | -40 |
| 10 - 13 | 16 | 11.5 | 0 | 0 | 0 |
| 13 - 16 | 4 | 14.5 | 3 | 1 | 4 |
| 16 - 19 | 4 | 17.5 | 6 | 2 | 8 |
| Total | Σfi = 100 | | | | Σfiui = -106 |
Now,
x = a + (Σf
iu
i/Σf
i) x h = 11.5 + (-106/100) x 3
= 11.5 - 318/100
= 11.5 - 3.18
= 8.32
Therefore, the mean of data is 8.32.
Here the maximum class frequency is 40, and the class corresponding to this frequency is 7 – 10.
So, the modal class is 7 – 10.
Now, Modal class = 7 – 10,
lower limit(l) of modal class = 7,
class size(h) = 3
frequency(f1) of modal class = 40,
frequency(f0) of modal class = 30,
frequency(f2) of modal class = 16,
Therefore, Mode = l + [(f1 - f0)/(2f1 - f0 - f2)] x h
= 7 + [(40-30)/{(2x40)-30-16}] x 3
= 7 + [10/(80-46)] x 3
= 7 + (10/34 x 3)
= 7 + 30/34
= 7 + 0.88
= 7.88
Therefore, the mode of this data is 7.88.
Hence, median and mean of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.88.
The distribution below gives the weights of 30 students of a class. Find the median weight of the students.
| Weights (in kg) | Number of students | Cumulative Frequency (cf) |
| 40 - 45 | 2 | 2 |
| 45 - 50 | 3 | 2 + 3 = 5 |
| 50 - 55 | 8 | 5 + 8 = 13 |
| 55 - 60 | 6 | 13 + 6 = 19 |
| 60 - 65 | 6 | 19 + 6 = 25 |
| 65 - 70 | 3 | 25 + 3 = 28 |
| 70 - 75 | 2 | 28 + 2 = 30 |
| Total | n = 30 | |
Now, n = 30
n/2 = 30/2 = 15 is 19. So, this observation lies in the class 55 - 60.
Then, l (the lower limit) = 55,
cf (the cumulative frequency of the class preceding 55 - 60) = 13,
f (the frequency of the median class 55 - 60) = 6,
h (the class size) = 5.
Therefore, Median = l + [(n/2 - cf)/f] x h
= 55 + [(30/2 - 13)/6] x 5
= 55 + [(15 - 13)/6] x 5
= 55 + (2/6 x 5)
= 55 + 10/6
= 55 + 1.67
= 56.67
Therefore, the median weight is 56.67 kg.
