The given data points are:
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6 partsStep-by-step solution
6 stepsTo solve this problem, we first need to organize the given data into a frequency distribution table using the specified class intervals (10-19, 20-29, etc.).
The given data points are:
23, 45, 67, 34, 56, 78, 89, 12, 45, 67,
54, 32, 76, 88, 43, 21, 65, 98, 54, 37,
29, 84, 71, 63, 55, 49, 91, 33, 27, 58,
62, 74, 39, 81, 47, 53, 69, 25, 72, 44,
36, 59, 83, 66, 41, 57, 73, 52, 68, 35
There are data points.
Here is the frequency distribution table:
| Class Interval | Class Mark (x) | Frequency (f) | fx | Cumulative Frequency (cf) |
|---|---|---|---|---|
| 10-19 | 14.5 | 1 | 14.5 | 1 |
| 20-29 | 24.5 | 5 | 122.5 | 6 |
| 30-39 | 34.5 | 7 | 241.5 | 13 |
| 40-49 | 44.5 | 7 | 311.5 | 20 |
| 50-59 | 54.5 | 9 | 490.5 | 29 |
| 60-69 | 64.5 | 8 | 516.0 | 37 |
| 70-79 | 74.5 | 6 | 447.0 | 43 |
| 80-89 | 84.5 | 5 | 422.5 | 48 |
| 90-99 | 94.5 | 2 | 189.0 | 50 |
| Total |
Part i· 1 step
Mean
Calculate the mean () using the formula .
The mean is .
Part ii
Median class
Determine the median position and identify the median class.
The median position is .
Looking at the cumulative frequency column, the 25th value falls within the class where the cumulative frequency first exceeds 25. This is the 50-59 class (cf = 29).
The median class is .
Part iii
Modal class
Identify the modal class.
The modal class is the class with the highest frequency. From the table, the highest frequency is 9, which corresponds to the 50-59 class.
The modal class is .
Part iv
Mean deviation
Calculate the mean deviation (MD) using the formula .
First, we calculate and for each class. .
| Class Mark (x) | | Frequency (f) | | | :------------- | :-------------- | :------------ | :--------------- | | 14.5 | 40.6 | 1 | 40.6 | | 24.5 | 30.6 | 5 | 153.0 | | 34.5 | 20.6 | 7 | 144.2 | | 44.5 | 10.6 | 7 | 74.2 | | 54.5 | 0.6 | 9 | 5.4 | | 64.5 | 9.4 | 8 | 75.2 | | 74.5 | 19.4 | 6 | 116.4 | | 84.5 | 29.4 | 5 | 147.0 | | 94.5 | 39.4 | 2 | 78.8 | | Total | | | |
The mean deviation is .
Part v
Semi-interquartile range
Calculate the semi-interquartile range (SIQR) using the formula .
First, find the first quartile () and third quartile ().
position: value. This falls in the 30-39 class.
position: value. This falls in the 70-79 class.The formula for a quartile is , where L is the lower class boundary, is the cumulative frequency before the quartile class, is the frequency of the quartile class, and c is the class width.
For :
, , , .For :
, , , .Now, calculate SIQR:
The semi-interquartile range is .
Part vi
Standard deviation
Calculate the standard deviation (s) using the formula .
First, we calculate and for each class. .
Class Mark (x) Frequency (f) 14.5 -40.6 1648.36 1 1648.36 24.5 -30.6 936.36 5 4681.80 34.5 -20.6 424.36 7 2970.52 44.5 -10.6 112.36 7 786.52 54.5 -0.6 0.36 9 3.24 64.5 9.4 88.36 8 706.88 74.5 19.4 376.36 6 2258.16 84.5 29.4 864.36 5 4321.80 94.5 39.4 1552.36 2 3104.72 Total The standard deviation is .
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