Standard Deviation

Q.

A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings $5.50mm$, $5.55mm$, $5.45mm$, $5.65mm$. The average of these four readings is $5.5375mm$ and the standard deviation of the data is $0.07395mm$. The average diameter of the pencil should therefore be recorded as:

1. $(5.54\pm 0.07)mm$

2. $(5.5375\pm 0.0740)mm$

3. $(5.5375\pm 0.0739)mm$

4. $(5.538\pm 0.074)mm$

Q.

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.

Q. The mean and standard deviation of $10$ observations are $20$ and $2$ respectively. Each of these $10$ observations is multiplied by $p$ and then reduced by $q$, (where $p\ne 0$ and $q\ne 0$). If the new mean and standard deviation becomes half of their original values, then the value of $pq$ is

Q.

Consider the numbers 1, 2, 34, 5, 67, 8, 9, 10. If 1 is added to each numbers, the variance of the numbers so obtained is

1. 6.5

2. 2.87

3. 3.87

4. 8.25

Q.

The standard deviation of first 10 natural numbers is

1. 2.87

2. 2.97

3. 5.5

4. 3.87

Q.

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, 321, 333, 348

(iv) 15, 22, 27, 11, 9, 21, 14, 9

Q. A data consists of $n$ observations:
$x_1,x_2,....,x_n$. If $\sum _{i=1}^n(x_i+1{)}^2=9n$ and $\sum _{i=1}^n(x_i-1{)}^2=5n$, then the standard deviation of this data is :

1. $\sqrt{7}$
2. $\sqrt{5}$
3. $5$
4. $2$

Q. The chance that a doctor will diagnose a certain disease correctly is $60\%$. The chance that a patient of a doctor will die by this treatment after correct diagnosis is $40\%$ and the chances of death by wrong diagnosis is $70\%$. The chances that the patient of a doctor having the particular disease will survive is $\frac{2K}{25}$. Then $K=$

Q.

If $r=0.5,\sum xy=120,\sum x^2=90,{\sigma }_y=8$, then $n=$

1. $100$

2. $10$

3. $15$

4. $50$

Q.

The standard deviation of $n$ observations $x_1,x_2,.......,x_n$ is $2$. If $\sum _{i=1}^n{x}_i=20$ and $\sum _{i=1}^n{x}_i^2=100$, then $n=$

1. $10$ or $20$

2. $5$ or $10$

3. $5$ or $20$

4. $5$ or $15$

5. $25$

Q.

Find the average of all odd numbers up to $100$.

Q.

Write the variance of first n natural numbers.

Q.

The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is

1. 50, 000

2. 250, 000

3. 255000

4. 252500

Q. A sample of 35 observations has the mean 80 and S.D as 4. A second sample of 65 observations from the same population has mean 70 and S.D. 3. The s.d. of the combined sample is

1. 5.85
2. 5.58
3. none of these
4. 3.42

Q.

Show that the two formulae for the standard deviation of ungrouped data .

$\sigma =\sqrt{\frac{1}{n}\sum (x_i-\overline{X}{)}^2}and{\sigma }^{\prime }=\sqrt{\frac{1}{n}\sum x_i^2-{\overline{X}}^2}$ are equivalent , where $\overline{X}=\frac{1}{n}\sum x_i$

Q.

The standard deviation of the observations 6, 5, 9, 13, 12, 8, 10 is

1. 6

2. $\sqrt{6}$

3. $\frac{52}{7}$

4. $\sqrt{\frac{52}{7}}$

Q.

The geometric mean of $1,2,2^2,.....,2^n$ is

1. $2^{\frac{n}{2}}$

2. $n^{\frac{n+1}{2}}$

3. $2^{\frac{n(n+1)}{2}}$

4. $2^{\frac{n-1}{2}}$

Q.

A student obtained the mean and standard deviation of 100 observations as 40 and 5.1 respectively. It was late found that one observation was wrongly copied as 50, the correct figure being 40. Find the correct mean and S. D.

Q. Standard deviation is always non negative.

1. False
2. True

Q.

The coefficient of correlation between $x$ and $y$ is $0.6$, then covariance is $16$. The standard deviation of $x$ is $4$, then the standard deviation of $y$ is

1. $5$

2. $10$

3. $20/3$

4. None of these

Q.

Bettys bite-size candies are packaged in bags.

The number of candies per bag is normally distributed, with a mean of $50$ candies and a standard deviation of $3$.

At a quality control checkpoint, a sample of bags is checked, and $4$ bags contain fewer than $47$ candies.

How many bags were probably taken as samples?

Q. All students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

1. variance
2. mean
3. median
4. mode

Q.

An urn contains $4$ white and $3$ red balls.

Three balls are drawn with replacement from this urn.

Then, the standard deviation of the number of red balls drawn is

1. $\frac{6}{7}$

2. $\frac{36}{49}$

3. $\frac{5}{7}$

4. $\frac{25}{49}$

Q. The mean and S.D. of $1,2,3,4,5,6$is

1. $3,3$
2. $\frac{7}{2},\sqrt{\frac{35}{12}}$
3. $\frac{7}{2},\sqrt{3}$
4. $\frac{7}{2},\frac{35}{12}$

Q.

The mean and 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.

Q.

The mean and standard deviation of marks obtained by 50 students of a class in three subjects, mathematics, physics and chemistry are given below :

$\begin{array}{cccc}\text{Subject}& \text{Mathematics}& \text{Physics}& \text{Chemistry}\\ \text{Mean}& 42& 32& 40.9\\ \text{Standard deviation}& 12& 15& 20\end{array}$

Which of the three subjects shows the highest variability in marks and which shows the lowest ?

Q.

If the sum of the squares of deviations for 10 observations taken from their mean is 2.5, then write thevalue of standard deviation.

Q. The second group of two samples has $100$ items with mean $15$ and S.D$=3$.If the whole group has $250$ items with mean $15.6$ and S.D$=\sqrt{13.44}$, then S.D of first group is

Q. The probability that an event $A$ occurs in a single trial of an experiment is $\mathrm{0.3.}$ Six independent trials of the experiment are performed. What is the variance of probability distribution of occurrence of event $A$?

1. $1.8$
2. $0.18$
3. $12.6$
4. $1.26$

Q.

While calculating the mean and variance of 10 readings, a student wrongly used the reading of 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.