Elementary Statistics MCQ Quiz - Objective Question with Answer for Elementary Statistics - Download Free PDF

Formula used:

Median class is found using the formula:

Median = L + [(N/2 - CF) / f] × h

Where:

L = lower boundary of the median class

N = total frequency

CF = cumulative frequency of the class before the median class

f = frequency of the median class

h = class width

Calculations:

Total frequency (N) = 8 + 16 + 36 + 34 + 6 = 100

N/2 = 100/2 = 50

Cumulative frequencies (CF):

CF for 0-10 = 8

CF for 10-20 = 8 + 16 = 24

CF for 20-30 = 24 + 36 = 60

CF for 30-40 = 60 + 34 = 94

CF for 40-50 = 94 + 6 = 100

The median class is the one in which the cumulative frequency exceeds N/2 (50). This occurs in the class interval 20-30, where CF = 24 and the next cumulative frequency is 60.

For the median class (20-30), the values are:

L = 20 (lower boundary of the median class)

f = 36 (frequency of the median class)

CF = 24 (cumulative frequency before the median class)

h = 10 (class width)

Now, applying the formula:

Median = 20 + [(50 - 24) / 36] × 10

⇒ Median = 20 + [26 / 36] × 10

⇒ Median = 20 + (0.7222 × 10)

⇒ Median = 20 + 7.222

⇒ Median = 27.222

∴ The median is 27.22

Given:

We are given a set of P values, and we need to find the algebraic sum of their deviations from their mean.

Formula used:

Algebraic sum of deviations = Σ(P_i - Mean)

Where, P_i represents the individual values in the set, and Mean is the average of the values.

Calculations:

Algebraic sum of deviations = Σ(P_i) - n × Mean

Where, n is the number of values in the set.

Since the Mean is calculated as:
Mean = Σ(P_i) / n, the expression becomes:

Algebraic sum of deviations = Σ(P_i) - n × (Σ(P_i) / n)

⇒ Algebraic sum of deviations = Σ(P_i) - Σ(P_i)

⇒ Algebraic sum of deviations = 0

∴ The algebraic sum of deviations from the mean is 0.

Given:

Variate values: 3, 4, 6, 7, 8, 14

Number of variate values (n) = 7

Formula Used:

Mean = (Sum of all variate values) / n

Sum of deviations from the mean = Σ(x_i - Mean) where x_i is each variate value

Calculation:

Sum of all variate values = 3 + 4 + 6 + 7 + 8 + 14

Sum of all variate values = 42

Mean = Sum of all variate values / n

Mean = 42 / 7

Mean = 6

Now calculate the deviations of each variate value from the mean:

(3 - 6) + (4 - 6) + (6 - 6) + (7 - 6) + (8 - 6) + (14 - 6)

= -3 + (-2) + 0 + 1 + 2 + 8

Sum of deviations = -3 - 2 + 0 + 1 + 2 + 8

Sum of deviations = 0

The sum of deviations from the mean is 0.

Calculation:

To find the median, we first need to arrange the data in ascending order:

1, 1, 3, 4, 4, 6, 7, 8, 9, 11

There are 10 data points (an even number). For an even number of data points, the median is the average of the two middle values.

The two middle values are the 5th and 6th values in the sorted list, which are 4 and 6.

Median = (4 + 6) / 2 = 10 / 2 = 5

Therefore, the median of the given data is 5.

Calculation:

Formula for Estimated Mean (x̄):

x̄ = Σ(f × x) / Σf

Calculation:

x̄ = 1288 / 21

x̄ ≈ 61.33

The estimated mean of the data is approximately 61.33.

Given:

If mode = 8 and mean – median = 12

Formula used

Mode = mean – 3 (mean - median)

Mode = 3median - 2mean

Calculation

We know that, Mode = mean – 3(mean -median)

Put the value, 8 = mean – 3 (12)

Mean = 36 + 8 = 44

Concept:

The mode is the value that appears most often in a set of data values.

Calculation:

32 occurred 8 times

14 occurred 4 times

59 occurred 12 times

41 occurred 8 times 

28 occurred 10 times

7 occurred 16 times 

34 occurred 15 times

20 occurred 9 times

∴ Mode will be 7

Concept:

Relation between mode, median and mean is given by:

Mode = 3 × median – 2 × mean

Calculation:

Given:

Mode – median = 2

As we know

Mode = 3 × median – 2 × mean

Now, Mode = median + 2

⇒ (2 + median) = 3median – 2mean   

⇒ 2Median - 2Mean = 2

⇒ Median - Mean = 1

Mean is the average of the given numbers,

⇒ Mean = (36 + 28 + 45 + 51)/4 = 160/4 = 40

Variance is calculated by taking the average of the squares of the difference between each term and the mean,

⇒ Variance = [(36 - 40)² + (28 - 40)² + (45 - 40)² + (51 - 40)²]/4

= [16 + 144 + 25 + 121]/4 = 306/4 = 76.5

Given:

Data is 3, 10, 10, 4, 7, 10, 5 

Formula used:

Average deviation about the mean 

\(∑\rm \frac{|x_{i} - x̅|}{n}\) where x̅ = Mean

xi = individual term 

n = total number of terms

Mean = Sum of all the terms/Total number of terms

Calculation:

n = total numbers in a data = 7

Mean x̅ = (3 + 10 + 10 + 4 + 7 + 10 + 5)/7 = 7

Mean deviation from mean = \(∑\rm \frac{|x_{i} - x̅|}{n}\)

Mean deviation from mean = (1/7) × [4 + 3 + 3 + 3 + 0 + 3 + 2]

∴ Mean deviation = 18/7

Given:

Mean of five consecutive even numbers = 16

Formula used:

\({\rm{V}} = \frac{{∑ {{\left| {{\rm{x}} - {\rm{m}}} \right|}^2}}}{{\rm{n}}}\)

\({\rm{Mean\;}}\left( {\rm{m}} \right) = \;\frac{{\left\{ {2{\rm{a\;}} + \left( {{\rm{n\;}} - 1} \right){\rm{d}}} \right\}}}{2}\)

V = variance

∑ = summation

x = observation

n = number of observations

a = 1^st term of the numbers

d = common difference

Calculation:

\(\frac{{\left\{ {2{\rm{a\;}} + \left( {{\rm{n\;}} - 1} \right){\rm{d}}} \right\}}}{2} = 16\)

⇒ 2a + (5 – 1)2 = 32

⇒ 2a + 4 × 2 = 32

⇒ 2a = 32 – 8

⇒ 2a = 24

⇒ a = 12

1^st term = 12

Other terms are 14, 16, 18, 20

\({\rm{V}} = {\rm{\;}}\frac{{{{\left( {12{\rm{\;}} - 16} \right)}^2} + {{\left( {14{\rm{\;}} - 16} \right)}^2} + {{\left( {16{\rm{\;}} - 16} \right)}^2} + {{\left( {18{\rm{\;}} - 16} \right)}^2} + {{\left( {20{\rm{\;}} - 16} \right)}^2}}}{5}\)

⇒ \({\rm{\;}}\frac{{16{\rm{\;}} + {\rm{\;}}4{\rm{\;}} + {\rm{\;}}0{\rm{\;}} + {\rm{\;}}4{\rm{\;}} + 16}}{5}\)

⇒ \({\rm{\;}}\frac{{40}}{5}\)

⇒ 8

⇒ V = 8

∴ The variance of the numbers is 8

Given

3, 4, 5, 7, 10, 10, 10

Concept used

Mean = Average

Deviation is the difference with the given number in the series.

Calculation

Mean = \(\frac{{3 + 4 + 5 + 7 + 10 + 10 + 10}}{7}\)

Mean = 49/7

Mean = 7

Checking the mean deviation with all the numbers given in the series.

Mean deviation 

⇒ |7 - 3|, |7 - 4|, |7 - 5|, |7 - 7|, |7 - 10|, |7 - 10|, |7 - 10|

⇒ 4, 3, 2, 0, 3, 3, 3

Mean deviation = \(\frac{{3 + 4 + 2 + 3 + 3 + 3}}{7}\)

Mean deviation = 18/7

Given:

The mid value of a class = 12

Width = 6

Formula used:

Lower limit = Mid value – width/2

Calculation:

Lower limit = 12 – 6/2

⇒ 12 – 3

⇒ 9

∴ The lower limit of the class is 9

GIVEN :

The standard deviation of a data set is given as 34.

CONCEPT :

The value of variance is the square of standard deviation.

FORMULA USED :
Standard Deviation = √Variance

CALCULATION :  

Using the formula :

Given:

7, 13, 15, 11, 4

Formula used:

 \({\rm{S}}.{\rm{D}} = √ {\frac{{∑|{\rm{x}} - {\rm{\;m|^2}}}}{{\rm{n}}}} \)

Mean (m) = Total of observations/number of observations

S.D = standard deviation

∑ = summation

x = observation

m = mean of the observations

n = number of observation

Calculation:

Mean of 7, 13, 15, 11, 4

⇒ 50/5

⇒ 10

\({\rm{S}}.{\rm{D}} = √ {\frac{{{{\left( {7 - 10} \right)}^2} + {{\left( {13 - 10} \right)}^2} + {{\left( {15\; - \;10} \right)}^2} + {{\left( {11 - 10} \right)}^2} + \;{{\left( {4 - 10} \right)}^2}}}{5}} \)

⇒ \(√ {\frac{{9 + \;9 + 25 + 1 + 36}}{5}} \)

⇒ \(√ {\frac{{80}}{5}} \)

⇒ √16

⇒ 4

∴ The standard deviation is 4