“Statistics may be rightly called the science of averages and their
estimates.” – A.L.BOWLEY & A.L. BODDINGTON
Introduction
We know that statistics deals with data collected for specific
purposes. We can make decisions about the data by
analysing and interpreting it. In earlier classes, we have
studied methods of representing data graphically and in
tabular form. This representation reveals certain salient
features or characteristics of the data. We have also studied
the methods of finding a representative value for the given
data. This value is called the measure of central tendency.
Recall mean (arithmetic mean), median and mode are three
measures of central tendency. A measure of central
tendency gives us a rough idea where data points are
centred. But, in order to make better interpretation from the
data, we should also have an idea how the data are scattered or how much they are
bunched around a measure of central tendency.
Consider now the runs scored by two batsmen in their last ten matches as follows:
Batsman A : 30, 91, 0, 64, 42, 80, 30, 5, 117, 71
Batsman B : 53, 46, 48, 50, 53, 53, 58, 60, 57, 52
Clearly, the mean and median of the data are
Batsman A Batsman B
Mean 53 53
Median 53 53
Recall that, we calculate the mean of a data (denoted by x ) by dividing the sum
of the observations by the number of observations .
Also, the median is obtained by first arranging the data in ascending or descending
order and applying the following rule. We find that the mean and median of the runs scored by both the batsmen A and
B are same i.e., 53.
Can we say that the performance of two players is same?
Clearly
No, because the variability in the scores of batsman A is from 0 (minimum) to 117
(maximum). Whereas, the range of the runs scored by batsman B is from 46 to 60.
Thus, the measures of central tendency are not sufficient to give complete
information about a given data. Variability is another factor which is required to be
studied under statistics. Like ‘measures of central tendency’ we want to have a
single number to describe variability. This single number is called a ‘measure of
dispersion’. In this Chapter, we shall learn some of the important measures of dispersion
and their methods of calculation for ungrouped and grouped data.
IMPORTANT QUESTIONS OF STATISTICS
1. Find the mean deviation about the mean for the following data:
6, 7, 10, 12, 13, 4, 8, 12
2. Find the mean deviation about the mean for the following data :
12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5
3. Find the mean deviation about the median for the following data:
3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21.
4. Find mean deviation about the mean for the following data :
xi 2 5 6 8 10 12
f
i 2 8 10 7 8 5
5. Find the mean deviation about the mean for the data in Exercises 1 and 2.
1. 4, 7, 8, 9, 10, 12, 13, 17
2. 38, 70, 48, 40, 42, 55, 63, 46, 54, 44
6. Find the mean deviation about the median for the data in Exercises 3 and 4.
3. 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
4. 36, 72, 46, 42, 60, 45, 53, 46, 51, 49
7. Coefficient of variation of two distributions are 60 and 70, and their
standard deviations are 21 and 16, respectively. What are their arithmetic means.
8. The variance of 20 observations is 5. If each observation is multiplied by
2, find the new variance of the resulting observations.
9. The mean of 5 observations is 4.4 and their variance is 8.24. If three of
the observations are 1, 2 and 6, find the other two observations.
10. If each of the observation x1
, x2
, ...,xn
is increased by ‘a’, where a is a
negative or positive number, show that the variance remains unchanged.
11. The mean and standard deviation of 100 observations were calculated as
40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one
observation. What are the correct mean and standard deviation?
12. The mean and variance of eight observations are 9 and 9.25, respectively. If six
of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
13. The mean and variance of 7 observations are 8 and 16, respectively. If five of the
observations are 2, 4, 10, 12, 14. Find the remaining two observations.
14. The mean and standard deviation of six observations are 8 and 4, respectively. If
each observation is multiplied by 3, find the new mean and new standard deviation
of the resulting observations.
15. Given that x is the mean and σ
2 is the variance of n observations x1
, x2
, ...,xn
.
Prove that the mean and variance of the observations ax1
, ax2
, ax3
, ...., axn
are
a x and a
2 σ
2
, respectively, (a ≠ 0).
16. The mean and standard deviation of 20 observations are found to be 10 and 2,
respectively. On rechecking, it was found that an observation 8 was incorrect.
Calculate the correct mean and standard deviation in each of the following cases:
(i) If wrong item is omitted. (ii) If it is replaced by 12.
17. The mean and standard deviation of a group of 100 observations were found to
be 20 and 3, respectively. Later on it was found that three observations were
incorrect, which were recorded as 21, 21 and 18. Find the mean and standard
deviation if the incorrect observations are omitted.
18. If the variance of a data is 121, then the standard deviation of the data is _______.
19. The standard deviation of a data is ___________ of any change in origin, but is
_____ on the change of scale.
20. The sum of the squares of the deviations of the values of the variable is _______
when taken about their arithmetic mean.
21. The mean deviation of the data is _______ when measured from the median.
22. The standard deviation is _______ to the mean deviation taken from the arithmetic
mean.
23. Mean and standard deviation of 100 observations were found to be 40 and 10,
respectively. If at the time of calculation two observations were wrongly taken
as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.
24. While calculating the mean and variance of 10 readings, a student wrongly used
the reading 52 for the correct reading 25. He obtained the mean and variance as
45 and 16 respectively. Find the correct mean and the variance.
25. Determine mean and standard deviation of first n terms of an A.P. whose first
term is a and common difference is d.
26. The mean life of a sample of 60 bulbs was 650 hours and the standard deviation
was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and
standard deviation 7 hours. Find the overall standard deviation.
27. Mean and standard deviation of 100 items are 50 and 4, respectively. Find the
sum of all the item and the sum of the squares of the items.
28. Two sets each of 20 observations, have the same standard derivation 5. The first
set has a mean 17 and the second a mean 22. Determine the standard deviation
of the set obtained by combining the given two sets.
29. Calculate the mean deviation about the mean of the set of first n natural numbers
when n is an odd number.
30. Calculate the mean deviation about the mean of the set of first n natural numbers
when n is an even number.
31. Find the standard deviation of the first n natural numbers.
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