Unit -5
Averages
Q1)- Briefly explain measures of Central Tendency?
A1)-
Measures of Central Tendency
A measure of central tendency is a statistical summary that represents the center point of the dataset. It indicates where most values in a distribution fall. It is also called as measure of central location.
The three most common measure of central tendency are Mean, Median, and Mode.
Definition
According to Prof Bowley, “Measures of central tendency (averages) are statistical constants which enable us to comprehend in a single effort the significance of the whole.”
Requisites of a good measure of central tendency
- It should be rigidly defined
- It should be simple to understand and easy to calculate
- It should be based upon all values of given data.
- It should be capable of further mathematical treatment.
- It should have sampling stability.
- It should be not be unduly affected by extreme values.
Mean
- The mean is the arithmetic average, also called as arithmetic mean.
- Mean is very simple to calculate and is most commonly used measure of the center of data.
- Means is calculated by adding up all the values and divided by the number of observation.
Merits of mean
- It is rigidly defined
- It is easy to understand and easy to calculate
- It is based upon all values of the given data
- It is capable of future mathematical treatment
- It is not much affected by sampling fluctuation
Demerits of mean
- It cannot be calculated if any observation are missing
- It cannot be calculated for open end classes
- It is effected by extreme values
- It cannot be located graphically
- It may be number which is not present in the data
Median
- The points or value that divides the data into two equal parts
- Firstly , the data are arranged in ascending or descending order .
- The median is the middle number depending on the data size.
- When the data size is odd, the median is the middle value
- When the data size is even, median is the average of the middle two values
- It is also known as middle score or 50th percentile
Merits of median
- It is rigidly defined
- It is easy to understand and easy to calculate
- It is not effected by extreme values
- It is not much affected by sampling fluctuation
- It can be located graphically
Demerits of median
- It is not based upon all values of the given data
- It is difficult to calculate increasing order data size
- It is not capable of further mathematical treatment.
Mode
The mode is denoted Mo, is the value which occurs most frequently in a set of values. Croxton and Cowden defined it as “the mode of a distribution is the value at the point armed with the item tends to most heavily concentrated. It may be regarded as the most typical of a series of value”
Merits of mode
- It is easy to understand & easy to calculate
- It is not affected by extreme values or sampling fluctuations.
- Even if extreme values are not known mode can be calculated.
- It can be located just by inspection in many cases.
- It is always present within the data.
Demerits of mode
- It is not rigidly defined.
- It is not based upon all values of the given data.
- It is not capable of further mathematical treatment.
Q2)- Find the mean
Xi | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Freq (Fi) | 2 | 5 | 12 | 17 | 14 | 6 | 3 |
A1) -
Xi | Freq (Fi) | XiFi |
9 | 2 | 18 |
10 | 5 | 50 |
11 | 12 | 132 |
12 | 17 | 204 |
13 | 14 | 182 |
14 | 6 | 84 |
15 | 3 | 45 |
| Fi = 59 | XiFi= 715 |
|
|
|
Then, N = ∑ fi = 59, and ∑fi Xi=715
X = 715/59 = 12.11
Q3) The following data represent the income distribution of 100 families. Calculate mean income of 100 families?
Income | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |
No. Of families | 8 | 12 | 25 | 22 | 16 | 11 | 6 |
A3)-
Income | No. Of families | Xm (Mid point) | FXm |
30-40 | 8 | 35 | 280 |
40-50 | 12 | 34 | 408 |
50-60 | 25 | 55 | 1375 |
60-70 | 22 | 65 | 1430 |
70-80 | 16 | 75 | 1200 |
80-90 | 11 | 85 | 935 |
90-100 | 6 | 95 | 570 |
| n = 100 |
| ∑f Xm = 6198 |
X = ∑f Xm/n = 6330/100 = 63.30
Mean = 63.30
Q4)- Calculate the mean number of hours per week spent by each student in texting message.
Time per week | 0 - 5 | 5 - 10 | 10 - 15 | 15 - 20 | 20 - 25 | 25 – 30 |
No. Of students | 8 | 11 | 15 | 12 | 9 | 5 |
A4)-
Time per week (X) | No. Of students (F) | Mid point X | XF |
0 - 5 | 8 | 2.5 | 20 |
5 – 10 | 11 | 7.5 | 82.5 |
10 - 15 | 15 | 12.5 | 187.5 |
15 - 20 | 12 | 17.5 | 210 |
20 - 25 | 9 | 22.5 | 202.5 |
25 – 30 | 5 | 27.5 | 137.5 |
| 60 |
| 840 |
Mean = 840/60 = 14
Q5)- The following table of grouped data represents the weights (in pounds) of all 100 babies born at a local hospital last year.
Weight (pounds) | Number of Babies |
[3−5) | 8 |
[5−7) | 25 |
[7−9) | 45 |
[9−11) | 18 |
[11−13) | 4 |
A5)-
Weight (pounds) | Number of Babies | Mid point X | XF |
[3−5) | 8 | 4 | 32 |
[5−7) | 25 | 6 | 150 |
[7−9) | 45 | 8 | 360 |
[9−11) | 18 | 10 | 180 |
[11−13) | 4 | 12 | 48 |
| 100 |
| 770 |
Mean = 770/100 = 7.7
Q6)- Find the median score of 7 students in science class
A6)-
Score = 19, 17, 16, 15, 12, 11, 10
Median = (7+1)/2 = 4th value
Median = 15
Q7)- Find the median score of 8 students in science class
A7)-
Score = 19, 17, 16, 15, 12, 11, 10, 9
Median = (8+1)/2 = 4.5th value
Median = (15+12)/2 = 13.5
Q8)- Find the median of the table given below
Marks obtained | No. Of students |
20 | 6 |
25 | 20 |
28 | 24 |
29 | 28 |
33 | 15 |
38 | 4 |
42 | 2 |
43 | 1 |
A8)-
Marks obtained | No. Of students | Cf |
20 | 6 | 6 |
25 | 20 | 26 (20+6) |
28 | 24 | 50 (26+24) |
29 | 28 | 78 |
33 | 15 | 93 |
38 | 4 | 97 |
42 | 2 | 99 |
43 | 1 | 100 |
Median = (n+1)/2 = 100+1/2 = 50.5
Median = (28+29)/2 = 28.5
Q9)- Calculate the median
Marks | No. Of students |
0-4 | 2 |
5-9 | 8 |
10-14 | 14 |
15-19 | 17 |
20-24 | 9 |
A9)-
Marks | No. Of students | CF |
0-4 | 2 | 2 |
5-9 | 8 | 10 |
10-14 | 14 | 24 |
15-19 | 17 | 41 |
20-24 | 9 | 50 |
| 50 |
|
n = 50
n = 50/2= 25
2
The category containing n/2 is 15 -19
Lb = 15
Cfp = 24
f = 17
Ci = 4
Median = 15 + 25-24 *4 = 15.23
17
Q10)-Given the below frequency table calculate median
X | 60 – 70 | 70 – 80 | 80- 90 | 90-100 |
F | 4 | 5 | 6 | 7 |
A10)-
X | F | CF |
60 - 70 | 4 | 4 |
70 - 80 | 5 | 9 |
80 - 90 | 6 | 15 |
90 - 100 | 7 | 22 |
n = 22
n = 22/2= 11
2
The category containing n+1/2 is 80 - 90
Lb = 80
Cfp = 9
f = 6
Ci = 10
Median = 80 + 11-9 *10 = 83.33
Q11)- Calculate the median of grouped data
Class interval | 1-3 | 3-5 | 5-7 | 7-9 | 9-11 | 11-13 |
Frequency | 4 | 12 | 13 | 19 | 7 | 5 |
A11)-
CI | F | CF |
1-3 | 4 | 4 |
3-5 | 12 | 16 |
5-7 | 13 | 29 |
7-9 | 19 | 48 |
9-11 | 7 | 55 |
11-13 | 5 | 60 |
n = 60
n = 60/2= 30
2
The category containing n+1/2 is 7-9
Lb = 7
Cfp = 29
f = 19
Ci = 2
Median = 7 + 30-29 *2 = 7.105
19
12. Find the mode of scores of section A
Scores = 25, 24, 24, 20, 17, 18, 10, 18, 9, 7
Solution – Mode is 24, 18 as both have occurred twice.
Q12) Find the mode
Seconds | Frequency |
51 - 55 | 2 |
56 - 60 | 7 |
61 - 65 | 8 |
66 - 70 | 4 |
A13)-
The group with the highest frequency is the modal group: - 61-65
D1 = 8-7 = 1
D2 = 8-4 = 4
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 61 + (65-61) 1 = 61+4 (1/5) = 61.8
1+4
Mode = 61.8
Q13)- In a class of 30 students marks obtained by students in science out of 50 is tabulated below. Calculate the mode of the given data.
Marks obtained | No. Of students |
10 -20 | 5 |
20 – 30 | 12 |
30 – 40 | 8 |
40 - 50 | 5 |
A14)-
The group with the highest frequency is the modal group: - 20 -30
D1 = 12 - 5 = 7
D2 = 12 - 8 = 4
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 20 + (30-20) 7 = 20+10 (7/11) = 26.36
7+4
Mode = 61.8
Q14)- Based on the group data below, find the mode
Time to travel to work | Frequency |
1 – 10 | 8 |
11 -20 | 14 |
21 – 30 | 12 |
31 – 40 | 9 |
41 - 50 | 7 |
A15)-
The group with the highest frequency is the modal group: - 11 - 20
D1 = 14 - 8 = 6
D2 = 14 - 12 = 2
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 11 + (20-11) 6 = 11+9 (6/8) = 17.75
6+2
16. Compute the mode from the following frequency distribution
CI | F |
70-71 | 2 |
68-69 | 2 |
66-67 | 3 |
64-65 | 4 |
62-63 | 6 |
60-61 | 7 |
58-59 | 5 |
Solution:
The group with the highest frequency is the modal group: - 60 - 61
D1 = 7 - 6 = 1
D2 = 7 - 5 = 2
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 60 + (61-60) 1 = 60+1 (1/3) 60.85
1+2
17. What is Geometric Mean? Explain with examples?
Answer - Geometric mean
Geometric mean is a type of mean or average, which indicates the central tendency of a set of numbers by using the product of their values.
Definition
The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.
For ungrouped data
Geometric Mean, GM = Antilog ∑logxi
N
Example 1 – find the G.M of the values
X | Log X |
45 | 1.653 |
60 | 1.778 |
48 | 1.681 |
65 | 1.813 |
Total | 6.925 |
GM = Antilog ∑logxi
N
= Antilog 6.925/4
= Antilog 1.73
= 53.82
For grouped data
Geometric Mean, GM = Antilog ∑ f logxi
N
Example 2 – calculate the geometric mean
X | f |
60 – 80 | 22 |
80 – 100 | 38 |
100 – 120 | 45 |
120 – 140 | 35 |
|
|
Solution
X | f | Mid X | Log X | f log X |
60 – 80 | 22 | 70 | 1.845 | 40.59 |
80 – 100 | 38 | 90 | 1.954 | 74.25 |
100 – 120 | 45 | 110 | 2.041 | 91.85 |
120 – 140 | 35 | 130 | 2.114 | 73.99 |
Total | 140 |
|
| 280.68 |
GM = Antilog ∑ f logxi
N
= antilog 280.68/140
= antilog 2.00
GM = 100
Example 3 – calculate geometric mean
Class | Frequency |
2-4 | 3 |
4-6 | 4 |
6-8 | 2 |
8-10 | 1 |
Solution
Class | Frequency | x | Log x | Flogx |
2-4 | 3 | 3 | 1.0986 | 3.2958 |
4-6 | 4 | 5 | 1.2875 | 6.4378 |
6-8 | 2 | 7 | 0.5559 | 3.8918 |
8-10 | 1 | 9 | 0.2441 | 2.1972 |
| 10 |
|
| 15.8226 |
GM = Antilog ∑ f logxi
N
= antilog 15.8226/10
= antilog 1.5823
GM = 4.866
18. What is Harmonic Mean? Explain with examples.
Answer - Harmonic mean
Harmonic mean is quotient of “number of the given values” and “sum of the reciprocals of the given values
For ungrouped data
Example 1 - Calculate the harmonic mean of the numbers 13.2, 14.2, 14.8, 15.2 and 16.1
Solution
X | 1/X |
13.2 | 0.0758 |
14.2 | 0.0704 |
14.8 | 0.0676 |
15.2 | 0.0658 |
16.1 | 0.0621 |
Total | 0.3147 |
H.M of X = 5/0.3147 = 15.88
Example 2 - Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ?
X | 1/X |
8 | 0.125 |
9 | 0.111 |
6 | 0.167 |
11 | 0.091 |
10 | 0.100 |
5 | 0.200 |
Total | 0.794 |
H.M of X = 6/0.794 = 7.560
For grouped data
Example 3 - Calculate the harmonic mean for the below data
Marks | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 | 90-99 |
F | 2 | 3 | 11 | 20 | 32 | 25 | 7 |
Solution
Marks | X | F | F/X |
30-39 | 34.5 | 2 | 0.0580 |
40-49 | 44.5 | 3 | 0.0674 |
50-59 | 54.4 | 11 | 0.2018 |
60-69 | 64.5 | 20 | 0.3101 |
70-79 | 74.5 | 32 | 0.4295 |
80-89 | 84.5 | 25 | 0.2959 |
90-99 | 94.5 | 7 | 0.0741 |
Total |
| 100 | 1.4368 |
HM = 100/1.4368 = 69.59
Example 4 – find the harmonic mean of the given class
Ages | 4 | 5 | 6 | 7 |
No. Of students | 6 | 4 | 10 | 9 |
Solution
X | F | f/x |
4 | 6 | 1.50 |
5 | 4 | 0.80 |
6 | 10 | 1.67 |
7 | 9 | 1.29 |
| 29.00 | 5.25 |
HM = 29/5.25 = 5.5
Example 5 – calculate harmonic mean
Class | Frequency |
2-4 | 3 |
4-6 | 4 |
6-8 | 2 |
8-10 | 1 |
Solution
Class | Frequency | x | f/x |
2-4 | 3 | 3 | 1 |
4-6 | 4 | 5 | 0.8 |
6-8 | 2 | 7 | 0.28 |
8-10 | 1 | 9 | 0.11 |
| 10 |
| 2.19 |
Harmonic mean = 10/2.19 = 4.55
Unit -5
Unit -5
Averages
Q1)- Briefly explain measures of Central Tendency?
A1)-
Measures of Central Tendency
A measure of central tendency is a statistical summary that represents the center point of the dataset. It indicates where most values in a distribution fall. It is also called as measure of central location.
The three most common measure of central tendency are Mean, Median, and Mode.
Definition
According to Prof Bowley, “Measures of central tendency (averages) are statistical constants which enable us to comprehend in a single effort the significance of the whole.”
Requisites of a good measure of central tendency
- It should be rigidly defined
- It should be simple to understand and easy to calculate
- It should be based upon all values of given data.
- It should be capable of further mathematical treatment.
- It should have sampling stability.
- It should be not be unduly affected by extreme values.
Mean
- The mean is the arithmetic average, also called as arithmetic mean.
- Mean is very simple to calculate and is most commonly used measure of the center of data.
- Means is calculated by adding up all the values and divided by the number of observation.
Merits of mean
- It is rigidly defined
- It is easy to understand and easy to calculate
- It is based upon all values of the given data
- It is capable of future mathematical treatment
- It is not much affected by sampling fluctuation
Demerits of mean
- It cannot be calculated if any observation are missing
- It cannot be calculated for open end classes
- It is effected by extreme values
- It cannot be located graphically
- It may be number which is not present in the data
Median
- The points or value that divides the data into two equal parts
- Firstly , the data are arranged in ascending or descending order .
- The median is the middle number depending on the data size.
- When the data size is odd, the median is the middle value
- When the data size is even, median is the average of the middle two values
- It is also known as middle score or 50th percentile
Merits of median
- It is rigidly defined
- It is easy to understand and easy to calculate
- It is not effected by extreme values
- It is not much affected by sampling fluctuation
- It can be located graphically
Demerits of median
- It is not based upon all values of the given data
- It is difficult to calculate increasing order data size
- It is not capable of further mathematical treatment.
Mode
The mode is denoted Mo, is the value which occurs most frequently in a set of values. Croxton and Cowden defined it as “the mode of a distribution is the value at the point armed with the item tends to most heavily concentrated. It may be regarded as the most typical of a series of value”
Merits of mode
- It is easy to understand & easy to calculate
- It is not affected by extreme values or sampling fluctuations.
- Even if extreme values are not known mode can be calculated.
- It can be located just by inspection in many cases.
- It is always present within the data.
Demerits of mode
- It is not rigidly defined.
- It is not based upon all values of the given data.
- It is not capable of further mathematical treatment.
Q2)- Find the mean
Xi | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Freq (Fi) | 2 | 5 | 12 | 17 | 14 | 6 | 3 |
A1) -
Xi | Freq (Fi) | XiFi |
9 | 2 | 18 |
10 | 5 | 50 |
11 | 12 | 132 |
12 | 17 | 204 |
13 | 14 | 182 |
14 | 6 | 84 |
15 | 3 | 45 |
| Fi = 59 | XiFi= 715 |
|
|
|
Then, N = ∑ fi = 59, and ∑fi Xi=715
X = 715/59 = 12.11
Q3) The following data represent the income distribution of 100 families. Calculate mean income of 100 families?
Income | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |
No. Of families | 8 | 12 | 25 | 22 | 16 | 11 | 6 |
A3)-
Income | No. Of families | Xm (Mid point) | FXm |
30-40 | 8 | 35 | 280 |
40-50 | 12 | 34 | 408 |
50-60 | 25 | 55 | 1375 |
60-70 | 22 | 65 | 1430 |
70-80 | 16 | 75 | 1200 |
80-90 | 11 | 85 | 935 |
90-100 | 6 | 95 | 570 |
| n = 100 |
| ∑f Xm = 6198 |
X = ∑f Xm/n = 6330/100 = 63.30
Mean = 63.30
Q4)- Calculate the mean number of hours per week spent by each student in texting message.
Time per week | 0 - 5 | 5 - 10 | 10 - 15 | 15 - 20 | 20 - 25 | 25 – 30 |
No. Of students | 8 | 11 | 15 | 12 | 9 | 5 |
A4)-
Time per week (X) | No. Of students (F) | Mid point X | XF |
0 - 5 | 8 | 2.5 | 20 |
5 – 10 | 11 | 7.5 | 82.5 |
10 - 15 | 15 | 12.5 | 187.5 |
15 - 20 | 12 | 17.5 | 210 |
20 - 25 | 9 | 22.5 | 202.5 |
25 – 30 | 5 | 27.5 | 137.5 |
| 60 |
| 840 |
Mean = 840/60 = 14
Q5)- The following table of grouped data represents the weights (in pounds) of all 100 babies born at a local hospital last year.
Weight (pounds) | Number of Babies |
[3−5) | 8 |
[5−7) | 25 |
[7−9) | 45 |
[9−11) | 18 |
[11−13) | 4 |
A5)-
Weight (pounds) | Number of Babies | Mid point X | XF |
[3−5) | 8 | 4 | 32 |
[5−7) | 25 | 6 | 150 |
[7−9) | 45 | 8 | 360 |
[9−11) | 18 | 10 | 180 |
[11−13) | 4 | 12 | 48 |
| 100 |
| 770 |
Mean = 770/100 = 7.7
Q6)- Find the median score of 7 students in science class
A6)-
Score = 19, 17, 16, 15, 12, 11, 10
Median = (7+1)/2 = 4th value
Median = 15
Q7)- Find the median score of 8 students in science class
A7)-
Score = 19, 17, 16, 15, 12, 11, 10, 9
Median = (8+1)/2 = 4.5th value
Median = (15+12)/2 = 13.5
Q8)- Find the median of the table given below
Marks obtained | No. Of students |
20 | 6 |
25 | 20 |
28 | 24 |
29 | 28 |
33 | 15 |
38 | 4 |
42 | 2 |
43 | 1 |
A8)-
Marks obtained | No. Of students | Cf |
20 | 6 | 6 |
25 | 20 | 26 (20+6) |
28 | 24 | 50 (26+24) |
29 | 28 | 78 |
33 | 15 | 93 |
38 | 4 | 97 |
42 | 2 | 99 |
43 | 1 | 100 |
Median = (n+1)/2 = 100+1/2 = 50.5
Median = (28+29)/2 = 28.5
Q9)- Calculate the median
Marks | No. Of students |
0-4 | 2 |
5-9 | 8 |
10-14 | 14 |
15-19 | 17 |
20-24 | 9 |
A9)-
Marks | No. Of students | CF |
0-4 | 2 | 2 |
5-9 | 8 | 10 |
10-14 | 14 | 24 |
15-19 | 17 | 41 |
20-24 | 9 | 50 |
| 50 |
|
n = 50
n = 50/2= 25
2
The category containing n/2 is 15 -19
Lb = 15
Cfp = 24
f = 17
Ci = 4
Median = 15 + 25-24 *4 = 15.23
17
Q10)-Given the below frequency table calculate median
X | 60 – 70 | 70 – 80 | 80- 90 | 90-100 |
F | 4 | 5 | 6 | 7 |
A10)-
X | F | CF |
60 - 70 | 4 | 4 |
70 - 80 | 5 | 9 |
80 - 90 | 6 | 15 |
90 - 100 | 7 | 22 |
n = 22
n = 22/2= 11
2
The category containing n+1/2 is 80 - 90
Lb = 80
Cfp = 9
f = 6
Ci = 10
Median = 80 + 11-9 *10 = 83.33
Q11)- Calculate the median of grouped data
Class interval | 1-3 | 3-5 | 5-7 | 7-9 | 9-11 | 11-13 |
Frequency | 4 | 12 | 13 | 19 | 7 | 5 |
A11)-
CI | F | CF |
1-3 | 4 | 4 |
3-5 | 12 | 16 |
5-7 | 13 | 29 |
7-9 | 19 | 48 |
9-11 | 7 | 55 |
11-13 | 5 | 60 |
n = 60
n = 60/2= 30
2
The category containing n+1/2 is 7-9
Lb = 7
Cfp = 29
f = 19
Ci = 2
Median = 7 + 30-29 *2 = 7.105
19
12. Find the mode of scores of section A
Scores = 25, 24, 24, 20, 17, 18, 10, 18, 9, 7
Solution – Mode is 24, 18 as both have occurred twice.
Q12) Find the mode
Seconds | Frequency |
51 - 55 | 2 |
56 - 60 | 7 |
61 - 65 | 8 |
66 - 70 | 4 |
A13)-
The group with the highest frequency is the modal group: - 61-65
D1 = 8-7 = 1
D2 = 8-4 = 4
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 61 + (65-61) 1 = 61+4 (1/5) = 61.8
1+4
Mode = 61.8
Q13)- In a class of 30 students marks obtained by students in science out of 50 is tabulated below. Calculate the mode of the given data.
Marks obtained | No. Of students |
10 -20 | 5 |
20 – 30 | 12 |
30 – 40 | 8 |
40 - 50 | 5 |
A14)-
The group with the highest frequency is the modal group: - 20 -30
D1 = 12 - 5 = 7
D2 = 12 - 8 = 4
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 20 + (30-20) 7 = 20+10 (7/11) = 26.36
7+4
Mode = 61.8
Q14)- Based on the group data below, find the mode
Time to travel to work | Frequency |
1 – 10 | 8 |
11 -20 | 14 |
21 – 30 | 12 |
31 – 40 | 9 |
41 - 50 | 7 |
A15)-
The group with the highest frequency is the modal group: - 11 - 20
D1 = 14 - 8 = 6
D2 = 14 - 12 = 2
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 11 + (20-11) 6 = 11+9 (6/8) = 17.75
6+2
16. Compute the mode from the following frequency distribution
CI | F |
70-71 | 2 |
68-69 | 2 |
66-67 | 3 |
64-65 | 4 |
62-63 | 6 |
60-61 | 7 |
58-59 | 5 |
Solution:
The group with the highest frequency is the modal group: - 60 - 61
D1 = 7 - 6 = 1
D2 = 7 - 5 = 2
Mode = L1 + (L2 – L1) d1
d1 +d2
mode = 60 + (61-60) 1 = 60+1 (1/3) 60.85
1+2
17. What is Geometric Mean? Explain with examples?
Answer - Geometric mean
Geometric mean is a type of mean or average, which indicates the central tendency of a set of numbers by using the product of their values.
Definition
The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.
For ungrouped data
Geometric Mean, GM = Antilog ∑logxi
N
Example 1 – find the G.M of the values
X | Log X |
45 | 1.653 |
60 | 1.778 |
48 | 1.681 |
65 | 1.813 |
Total | 6.925 |
GM = Antilog ∑logxi
N
= Antilog 6.925/4
= Antilog 1.73
= 53.82
For grouped data
Geometric Mean, GM = Antilog ∑ f logxi
N
Example 2 – calculate the geometric mean
X | f |
60 – 80 | 22 |
80 – 100 | 38 |
100 – 120 | 45 |
120 – 140 | 35 |
|
|
Solution
X | f | Mid X | Log X | f log X |
60 – 80 | 22 | 70 | 1.845 | 40.59 |
80 – 100 | 38 | 90 | 1.954 | 74.25 |
100 – 120 | 45 | 110 | 2.041 | 91.85 |
120 – 140 | 35 | 130 | 2.114 | 73.99 |
Total | 140 |
|
| 280.68 |
GM = Antilog ∑ f logxi
N
= antilog 280.68/140
= antilog 2.00
GM = 100
Example 3 – calculate geometric mean
Class | Frequency |
2-4 | 3 |
4-6 | 4 |
6-8 | 2 |
8-10 | 1 |
Solution
Class | Frequency | x | Log x | Flogx |
2-4 | 3 | 3 | 1.0986 | 3.2958 |
4-6 | 4 | 5 | 1.2875 | 6.4378 |
6-8 | 2 | 7 | 0.5559 | 3.8918 |
8-10 | 1 | 9 | 0.2441 | 2.1972 |
| 10 |
|
| 15.8226 |
GM = Antilog ∑ f logxi
N
= antilog 15.8226/10
= antilog 1.5823
GM = 4.866
18. What is Harmonic Mean? Explain with examples.
Answer - Harmonic mean
Harmonic mean is quotient of “number of the given values” and “sum of the reciprocals of the given values
For ungrouped data
Example 1 - Calculate the harmonic mean of the numbers 13.2, 14.2, 14.8, 15.2 and 16.1
Solution
X | 1/X |
13.2 | 0.0758 |
14.2 | 0.0704 |
14.8 | 0.0676 |
15.2 | 0.0658 |
16.1 | 0.0621 |
Total | 0.3147 |
H.M of X = 5/0.3147 = 15.88
Example 2 - Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ?
X | 1/X |
8 | 0.125 |
9 | 0.111 |
6 | 0.167 |
11 | 0.091 |
10 | 0.100 |
5 | 0.200 |
Total | 0.794 |
H.M of X = 6/0.794 = 7.560
For grouped data
Example 3 - Calculate the harmonic mean for the below data
Marks | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 | 90-99 |
F | 2 | 3 | 11 | 20 | 32 | 25 | 7 |
Solution
Marks | X | F | F/X |
30-39 | 34.5 | 2 | 0.0580 |
40-49 | 44.5 | 3 | 0.0674 |
50-59 | 54.4 | 11 | 0.2018 |
60-69 | 64.5 | 20 | 0.3101 |
70-79 | 74.5 | 32 | 0.4295 |
80-89 | 84.5 | 25 | 0.2959 |
90-99 | 94.5 | 7 | 0.0741 |
Total |
| 100 | 1.4368 |
HM = 100/1.4368 = 69.59
Example 4 – find the harmonic mean of the given class
Ages | 4 | 5 | 6 | 7 |
No. Of students | 6 | 4 | 10 | 9 |
Solution
X | F | f/x |
4 | 6 | 1.50 |
5 | 4 | 0.80 |
6 | 10 | 1.67 |
7 | 9 | 1.29 |
| 29.00 | 5.25 |
HM = 29/5.25 = 5.5
Example 5 – calculate harmonic mean
Class | Frequency |
2-4 | 3 |
4-6 | 4 |
6-8 | 2 |
8-10 | 1 |
Solution
Class | Frequency | x | f/x |
2-4 | 3 | 3 | 1 |
4-6 | 4 | 5 | 0.8 |
6-8 | 2 | 7 | 0.28 |
8-10 | 1 | 9 | 0.11 |
| 10 |
| 2.19 |
Harmonic mean = 10/2.19 = 4.55
