In an experiment with 15 observations...

MathematicsMeasures of DispersionJEE Main 2003Easy

View in:English Hindi

In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

(A)

8.33

(B)

78.00

(C)

188.66

(D)

177.33

Answer:B

Visualized Solution (Hindi)

Identify Given Parameters

Number of observations $n = 15$
Given incorrect sum $\sum x = 170$
Given incorrect sum of squares $\sum x^{2} = 2830$

The Error Detection

Incorrect value $= 20$
Correct value $= 30$

Variance Formula Bridge

Variance formula: $σ^{2} = \frac{\sum x ^{2}}{n} - (\frac{\sum x}{n})^{2}$
We need corrected values for $\sum x$ and $\sum x^{2}$ .

Correcting the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x

Corrected $\sum x = (Old \sum x) - (Wrong Value) + (Correct Value)$
Corrected $\sum x = 170 - 20 + 30 = 180$

Correcting Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x^{2}

Corrected $\sum x^{2} = (Old \sum x^{2}) - (Wrong Value)^{2} + (Correct Value)^{2}$
Corrected $\sum x^{2} = 2830 - 2 0^{2} + 3 0^{2}$

Calculating Corrected .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x^{2}

Corrected $\sum x^{2} = 2830 - 400 + 900$
Corrected $\sum x^{2} = 3330$

Calculating Corrected Mean

Corrected Mean $\overset{x}{ˉ} = \frac{Corrected \sum x}{n}$
$\overset{x}{ˉ} = \frac{180}{15} = 12$

Final Variance Setup

Corrected Variance $= \frac{3330}{15} - (12)^{2}$

Execution of Division

$\frac{3330}{15} = 222$
$1 2^{2} = 144$

The Final Result

Corrected Variance $= 222 - 144$
Corrected Variance $= 78$

Key Takeaway

Key Takeaway: Always correct $\sum x$ and $\sum x^{2}$ separately before calculating variance.
Next Challenge: Try solving this if two observations were incorrect instead of one!

00:00 / 00:00

Conceptually Similar Problems

MathematicsMeasures of DispersionJEE Advanced 1979Easy

View in:English Hindi

In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.

Answer:Correct mean 42.3 ; Correct variance 43.81

MathematicsVariance and Standard DeviationJEE Advanced 1982Moderate

View in:English Hindi

The marks obtained by 40 students are grouped in a frequency table in class intervals of 10 marks each. The mean and the variance obtained from this distribution are found to be 40 and 49 respectively. It was later discovered that two observations belonging to the class interval (21-30) were included in the class interval (31-40) by mistake. Find the mean and the variance after correcting the error.

Answer:new mean = 39.5, new variance = 49.25

MathematicsMeasures of Central Tendency (Mean, Median, Mode)JEE Main 2015Easy

View in:English Hindi

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

(A)

15.8

(B)

14.0

(C)

16.8

(D)

16.0

Answer:B

MathematicsVariance and Standard DeviationJEE Advanced 1981Moderate

View in:English Hindi

The mean square deviations of a set of observations $x_{1}, x_{2}, \dots, x_{n}$ about a point $c$ is defined to be $\frac{1}{n} \sum_{i = 1}^{n} (x_{i} - c)^{2}$ . The mean square deviations about $- 1$ and $+ 1$ of a set of observations are 7 and 3 respectively. Find the standard deviation of this set of observations.

Answer:

3

MathematicsVariance and Standard DeviationJEE Main 2012Easy

View in:English Hindi

Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations, and let $\overset{x}{ˉ}$ be their arithmetic mean and $σ^{2}$ be the variance. Statement-1 : Variance of $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 σ^{2}$ Statement-2 : Arithmetic mean $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 \overset{x}{ˉ}$ .

(A)

Statement-1 is false, Statement-2 is true.

(B)

Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.

(C)

Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.

(D)

Statement-1 is true, statement-2 is false.

Answer:D

MathematicsMeasures of DispersionJEE Main 2006Easy

View in:English Hindi

Suppose a population A has 100 observations 101, 102, ..., 200 and another population B has 100 observations 151, 152, ..., 250. If $V_{A}$ and $V_{B}$ represent the variances of the two populations, respectively then $\frac{V _{A}}{V _{B}}$ is

(A)

(B)

\frac{9}{4}

(C)

\frac{4}{9}

(D)

\frac{2}{3}

Answer:A

MathematicsBinomial DistributionJEE Main 2002Easy

View in:English Hindi

A dice is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of success is

(A)

8/3

(B)

3/8

(C)

4/5

(D)

5/4

Answer:D

MathematicsMeasures of DispersionJEE Advanced 1981Easy

View in:English Hindi

Consider any set of 201 observations $x_{1}, x_{2}, \dots, x_{200}, x_{201}$ . It is given that $x_{1} < x_{2} < \dots < x_{200} < x_{201}$ . Then the mean deviation of this set of observations about a point $k$ is minimum when $k$ equals

(A)

(x_{1} + x_{2} + \dots + x_{200} + x_{201}) /201

(B)

x_{1}

(C)

x_{101}

(D)

x_{201}

Answer:C

MathematicsMeasures of DispersionJEE Main 2005Easy

View in:English Hindi

Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations such that $\sum x_{i}^{2} = 400$ and $\sum x_{i} = 80$ . Then the possible value of n among the following is

(A)

(B)

(C)

(D)

Answer:B

MathematicsMeasures of DispersionJEE Advanced 1980Easy

View in:English Hindi

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $c$ . Then the standard deviation of the resulting number is

(A)

s

(B)

cs

(C)

s c

(D)

none of these

Answer:B