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

Given:

The observations are: 4, 3, 8, 7, 3, 7, 3, 1, 1, 3, 8, 3, 3, 5, and 3.

Formula used:

Mode = The value that appears most frequently in a data set.

Calculations:

Frequency of each observation:

4 → 1 time

3 → 7 times

8 → 2 times

7 → 2 times

1 → 2 times

5 → 1 time

⇒ The observation that appears most frequently is 3 (7 times).

∴ The correct answer is option (2).

Given:

Mode = 89.7

Median = 32

Formula used:

Empirical formula (relationship between Mean, Median, and Mode):

Mode \(\approx\) 3 Median - 2 Mean

Calculations:

Rearrange the empirical formula to find the Mean:

2 Mean \(\approx\) 3 Median - Mode

Mean \(\approx \frac{(3 \times Median) - Mode}{2}\)

Substitute the given values:

Mean \(\approx \frac{(3 \times 32) - 89.7}{2} \)

⇒ Mean \(\approx \frac{96 - 89.7}{2}\)

⇒ Mean \(\approx \frac{6.3}{2}\)

⇒ Mean \(\approx 3.15\)

∴ The mean of the data set is approximately 3.15.

Given:

Observations = 28, 31, 40, 63, 57, 37, 34, 70, 99

Formula used:

Arithmetic Mean (AM) = (Sum of Observations) / (Number of Observations)

Calculation:

Sum of Observations = 28 + 31 + 40 + 63 + 57 + 37 + 34 + 70 + 99

⇒ Sum = 459

Number of Observations = 9

⇒ AM = 459 / 9

⇒ AM = 51

∴ The correct answer is option (4).

Given:

Mode = 52.7

Median = 65

Formula Used:

Empirical formula: Mean - Mode = 3 × (Mean - Median)

Calculations:

Mean - 52.7 = 3 × (Mean - 65)

⇒ Mean - 52.7 = 3 × Mean - 3 × 65

⇒ Mean - 52.7 = 3 × Mean - 195

⇒ 195 - 52.7 = 3 × Mean - Mean

⇒ 142.3 = 2 × Mean

⇒ Mean = 142.3 / 2

⇒ Mean = 71.15

⇒ Mean ≈ 71.2

The mean of the dataset is 71.2.

Given:

Marks = [19, 36, 60, 69, 85]

No. of Students = [63, 62, 59, 17, 70]

Formula Used:

Mean = (Σ(f_i × x_i)) / Σ(f_i)

Where:

f_i = Frequency (No. of Students)

x_i = Marks

Calculation:

f_i × x_i:

19 × 63 = 1197

36 × 62 = 2232

60 × 59 = 3540

69 × 17 = 1173

85 × 70 = 5950

Σ(f_i × x_i) = 1197 + 2232 + 3540 + 1173 + 5950 = 14092

Σ(f_i) = 63 + 62 + 59 + 17 + 70 = 271

Mean:

⇒ Mean = Σ(f_i × x_i) / Σ(f_i)

⇒ Mean = 14092 / 271

⇒ Mean = 52

The mean of the distribution is 52.

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