Question 1: Find the mean, variance and standard deviation for the following data:

(i) 2, 4, 5, 6, 8, 17 (ii) 6, 7, 10, 12, 13, 4, 8, 12

(iii) 227, 235, 255, 269, 292, 299, 312, 333, 348 (iv) 15, 22, 27, 11, 9, 21, 14, 9

Answer:

(i) Given data : 2, 4, 5, 6, 8, 17

Let be the mean of the given set of observations. Then,

*Computation of Variance*

(ii) Given data : 6, 7, 10, 12, 13, 4, 8, 12

Let be the mean of the given set of observations. Then,

*Computation of Variance*

(iii) Given data : 227, 235, 255, 269, 292, 299, 312, 333, 348

Let be the mean of the given set of observations. Then,

*Computation of Variance*

(iv) Given data : 15, 22, 27, 11, 9, 21, 14, 9

Let be the mean of the given set of observations. Then,

*Computation of Variance*

Question 2: The variance of 20 observations is 5. If each observation is multiplied by 2, find the variance of the resulting observations.

Answer:

Squaring both sides

Therefore the variance of the new observations is 20.

Question 3: The variance of 15 observations is 4. If each observation is increased by 9, find the variance of the resulting observations.

Answer:

Squaring both sides

Therefore the variance of the new observations is 4.

Question 4: 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 two other observations.

Answer:

Let and be the other two observations.

Solving i) and iv) we get and

Solving i) and v) we get and

Hence the other two observations are 4 and 9.

Question 5: The mean and standard deviation of 6 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.

Answer:

Therefore the new mean and new standard deviation of the resulting observations are 24 and 12respectively.

Question 6: The mean and the variance of 8 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.

Answer:

Let and be the other two observations.

Solving i) and iv) we get and

Solving i) and v) we get and

Hence the other two observations are 4 and 9.

Question 7: For a group of 200 candidates, the mean and standard deviations of scores were found to be 40 and 15 respectively. Later on it was discovered that the scores of 43 and 35 were misread as 34 and 53 respectively. Find the correct mean and standard deviation.

Answer:

Since the scores were misread, this sum is incorrect. The correct sum would be

We know,

Question 8: 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?

Answer:

Since the score was misread, this sum is incorrect. The correct sum would be

We know,

Question 9: 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

Answer:

Variance

(i) If wrong item is omitted

We know,

(ii) if it is replaced by 12

We know,

Question 10: The mean and standard deviation of 20 observations are 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 were omitted.

Answer:

Variance

When the incorrect observations 21, 12, 18 are removed, n = 97

We know,

Question 11: Show that the two formulae for the standard deviation of ungrouped date

Answer:

Hence