WEEK ELEVEN
TOPIC: Mean Deviation, Variance and standard Deviation of Grouped Data use in solving practical problems related to real life situations
Mean Deviation of Grouped Data
Example 1
The speeds of 40 cars in a certain road are tabulated as follows:
| Speed (km/h) | 50 – 54 | 55 – 59 | 60 – 64 | 65 – 69 | 60 – 74 | 75 – 80 | 80 – 84 |
| Frequency | 5 | 10 | 15 | 12 | 10 | 6 | 2 |
For this distribution, calculate
- The mean
- The mean deviation
Solution
The complete table of the distribution is shown below.
| Class interval | Mid – value (xm) | ||||
| 50 – 54 | 52 | 5 | 260 | 13.17 | 65.85 |
| 55 – 59 | 57 | 10 | 570 | 8.17 | 81.5 |
| 60 – 64 | 62 | 15 | 930 | 3.17 | 47.55 |
| 65 – 69 | 67 | 12 | 804 | 1.83 | 21.96 |
| 60 – 74 | 72 | 10 | 720 | 6.83 | 68.3 |
| 75 – 80 | 77 | 6 | 462 | 11.83 | 70.98 |
| 80 – 84 | 82 | 2 | 164 | 16.83 | 33.66 |
| Total |
- Mean, =
The mean is 65.2km/h to 1 d.p.
- Mean deviation =
The mean deviation is 6.5km/h
EVALUATION
- Calculate the mean and the mean deviation of the following:
- 8, 5, 12, 8, 13, 4, 9, 5, 4, 7
- 9.25, 8.04, 12.08, 9.82, 10.05, 2.05, 8.25, 7.64, 7.02, 8.02
Variance and Standard Deviation of a Grouped Data
Example 1
The table shows the time to the nearest hours of television watched by a group of students in a week.
| Time | 1 – 5 | 6 – 10 | 11 – 15 | 16 – 20 | 21 – 25 | 26 – 30 | 31 – 35 | 36 – 40 |
| Frequency | 2 | 5 | 8 | 10 | 14 | 6 | 4 | 1 |
Calculate
- The mean
- The variance
- The standard deviation
Solution
Let xm represents the mid-value (or class mark) of the interval.
- =
Now subtract 19.8 from each value in the 2nd column to obtain the results in the 5th column. Then complete the other two columns as shown in the table.
- S2 =
Variance = 64.8h to 3 s.f.
- S = = 8.047h
Standard deviation is 8.05h to 3 s.f.
Alternative method
| Interval | |||||
| 1 – 5 | 3 | 2 | 6 | 9 | 18 |
| 6 – 10 | 8 | 5 | 40 | 64 | 320 |
| 11 – 15 | 13 | 8 | 104 | 169 | 1352 |
| 16 – 20 | 18 | 10 | 180 | 324 | 3240 |
| 21 – 25 | 23 | 14 | 322 | 529 | 7406 |
| 26 – 30 | 28 | 6 | 168 | 784 | 4704 |
| 31 – 35 | 33 | 4 | 132 | 1089 | 4356 |
| 36 – 40 | 38 | 1 | 38 | 1444 | 1444 |
| Total |
EVALUATION
- Calculate to 1 d.p the mean and standard deviation of the following numbers:
- 5, 7, 12, 10, 5, 15, 14, 9, 7, 8
- 6.5, 8.5, 6.5, 8.4, 6.9, 2.5, 6.2, 5.5
GENERAL EVALUATION
- The table bellows shows the age distributions of a group of people.
| Age (yrs) | 20 – 29 | 30 – 39 | 40 – 49 | 50 – 59 | 60 – 69 | 70 – 79 |
| Frequency | 3 | 5 | 10 | 13 | 7 | 2 |
Calculate:
- The mean age
- The variance
- The standard deviation
READING ASSIGNMENT
Essential Mathematics for Senior Secondary 1 pgs 237 – 248
WEEKEND ASSIGNMENT
- The lowest temperatures of a city in Asia for 10 consecutive days are recorded as: – 5oC, – 6oC, -5oC, 4oC, 0oC, 1oC, 2oC, 3oC, 4oC, 7oC. Find the mean deviation. A. 3.9 B. 4.0 C. 3.6 D. 6.4
Use the table below to answer question 2 to 4
A dice is thrown 100 times. The results are recorded as shown in the following table
| Score | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 15 | 18 | 17 | 21 | 14 | 15 |
Calculate:
- The mean score A. 4.0 B. 3.5 C. 1.0 D. 5.6
- The variance A. 2.7 B. 3.7 C. 2.1 D. 1
- The standard deviation A. 4 B. 5.1 C. 1.6 D. 7
- Find the variance of x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x and 10x. A. 8.25x29x2 B. 10x2 7.25x2
THEORY
- The shoe sizes of a group of people are as follows:
| Shoe size | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Frequency | 3 | 8 | 14 | 16 | 20 | 10 | 5 | 3 | 1 |
For this distribution, calculate the mean deviation
- The table below show the age distributions of a group of people.
| Age (yrs) | 20 – 29 | 30 – 39 | 40 – 49 | 50 – 59 | 60 -69 | 70 – 79 |
| Frequency | 3 | 5 | 10 | 13 | 7 | 2 |
Calculate (a) the mean age (b) the variance (c) the standard deviation