Elementary Statistics MCQ Quiz - Objective Question with Answer for Elementary Statistics - Download Free PDF
Last updated on Jun 6, 2025
Latest Elementary Statistics MCQ Objective Questions
Elementary Statistics Question 1:
Find the median of the given data: 7, 9, 3, 4, 11, 1, 8, 6, 1, 4.
Answer (Detailed Solution Below)
Elementary Statistics Question 1 Detailed Solution
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.
Elementary Statistics Question 2:
Find out the estimated mean of the data in the following table
Answer (Detailed Solution Below)
Elementary Statistics Question 2 Detailed Solution
Calculation:
Seconds (Class Interval) | Frequency (f) | Midpoint (x) | f × x |
---|---|---|---|
51 - 55 | 2 | (51 + 55) / 2 = 53 | 2 × 53 = 106 |
56 - 60 | 7 | (56 + 60) / 2 = 58 | 7 × 58 = 406 |
61 - 65 | 8 | (61 + 65) / 2 = 63 | 8 × 63 = 504 |
66 - 70 | 4 | (66 + 70) / 2 = 68 | 4 × 68 = 272 |
Total | Σf = 21 | Σ(f × x) = 1288 |
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.
Elementary Statistics Question 3:
Find the mode of the following data:
59, 65, 61, 62, 53, 55, 60, 70, 64, 56, 58, 58, 62, 62, 68, 65, 56, 59, 68, 61, 67
Answer (Detailed Solution Below)
Elementary Statistics Question 3 Detailed Solution
Calculation:
To find the mode, we need to identify the number that appears most frequently in the data set.
Let's list the data and count the occurrences of each number:
53: 1
55: 1
56: 2
58: 2
59: 2
60: 1
61: 2
62: 3
64: 1
65: 2
67: 1
68: 2
70: 1
From the counts, the number 62 appears 3 times, which is more than any other number.
Therefore, the mode of the given data is 62.
Elementary Statistics Question 4:
The following question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following:
(a) Both A and R are true and R is a correct explanation of A
(b) Both A and R are true and R is not a correct explanation of A
(c) A is true and R is false
(d) A is false and R is true
Assertion (A) | Reason (R) |
---|---|
The mode of the data: 5, 9, 11, 7, 12, 18, 32, 39, 19, 52, 27, 21, 23, 52, 18, 34, 41, 18, 40, 17 is 52 | Mode is that value of variable in the data which has maximum frequency |
Answer (Detailed Solution Below)
Elementary Statistics Question 4 Detailed Solution
Given:
Assertion (A): The mode of the data: 5, 9, 11, 7, 12, 18, 32, 39, 19, 52, 27, 21, 23, 52, 18, 34, 41, 18, 40, 17 is 52.
Reason (R): Mode is that value of the variable in the data which has maximum frequency.
Analysis:
The data set is: 5, 9, 11, 7, 12, 18, 32, 39, 19, 52, 27, 21, 23, 52, 18, 34, 41, 18, 40, 17.
The frequencies of the numbers are as follows:
- 5: 1 time
- 9: 1 time
- 11: 1 time
- 7: 1 time
- 12: 1 time
- 18: 3 times
- 32: 1 time
- 39: 1 time
- 19: 1 time
- 52: 2 times
- 27: 1 time
- 21: 1 time
- 23: 1 time
- 34: 1 time
- 41: 1 time
- 40: 1 time
- 17: 1 time
Clearly, the number 18 occurs 3 times, which is more frequent than any other number in the data set. Therefore, the mode is 18, not 52.
The reason (R) is correct because the mode is indeed the value that occurs most frequently in a data set. However, the assertion (A) is false because the mode is not 52, but 18.
Conclusion:
The correct answer is (d) A is false and R is true.
Elementary Statistics Question 5:
If the mode of the following distribution is 3.6, then what is the value of (2k + 1)?
Class | 1–3 | 3–5 | 5–7 | 7–9 | 9–11 |
---|---|---|---|---|---|
Frequency | 16 | 22 | k | 11 | 7 |
Answer (Detailed Solution Below)
Elementary Statistics Question 5 Detailed Solution
Formula used:
The formula for the mode of a grouped frequency distribution is:
Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × h
Where: L is the lower boundary of the modal class,
f1 is the frequency of the modal class,
f0 is the frequency of the class preceding the modal class,
f2 is the frequency of the class succeeding the modal class,
h is the class width.
Calculation:
The mode is given as 3.6, so the modal class is 3–5 (since the mode lies between 3 and 5). In this class:
L = 3 (lower boundary of the class),
f1 = 22 (frequency of the modal class),
f0 = 16 (frequency of the class preceding the modal class),
f2 = k (frequency of the class succeeding the modal class),
h = 2 (class width).
We are given that the mode = 3.6, so substituting the values into the mode formula:
3.6 = 3 + [(22 - 16) / (2 × 22 - 16 - k)] × 2
3.6 = 3 + [6 / (44 - 16 - k)] × 2
3.6 = 3 + [6 / (28 - k)] × 2
3.6 - 3 = [6 / (28 - k)] × 2
0.6 = [6 / (28 - k)] × 2
0.6 / 2 = 6 / (28 - k)
0.3 = 6 / (28 - k)
0.3 × (28 - k) = 6
28 - k = 6 / 0.3
28 - k = 20
k = 28 - 20 = 8
Substitute k = 8 into (2k + 1):
(2k + 1) = 2 × 8 + 1 = 16 + 1 = 17
∴ The value of (2k + 1) is 17.
Top Elementary Statistics MCQ Objective Questions
If Mode is 8 and mean – median = 12 then find the value of mean?
Answer (Detailed Solution Below)
Elementary Statistics Question 6 Detailed Solution
Download Solution PDFGiven:
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
If the difference between the mode and median is 2, then find the difference between the median and mean(in the given order).
Answer (Detailed Solution Below)
Elementary Statistics Question 7 Detailed Solution
Download Solution PDFConcept:
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
∴ The difference between the median and mean is 1.What is the Mode of the following data:
X |
32 |
14 |
59 |
41 |
28 |
7 |
34 |
20 |
f(x) |
8 |
4 |
12 |
8 |
10 |
16 |
15 |
9 |
Answer (Detailed Solution Below)
Elementary Statistics Question 8 Detailed Solution
Download Solution PDFConcept:
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
Find the variance of the given numbers: 36, 28, 45, and 51.
Answer (Detailed Solution Below)
Elementary Statistics Question 9 Detailed Solution
Download Solution PDFMean 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
∴ Variance of the given numbers = 76.5The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from mean is :
Answer (Detailed Solution Below)
Elementary Statistics Question 10 Detailed Solution
Download Solution PDFGiven:
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
Mean of five consecutive even numbers is 16, find the variance of the numbers.
Answer (Detailed Solution Below)
Elementary Statistics Question 11 Detailed Solution
Download Solution PDFGiven:
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 = 1st 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
1st 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
Find the mean deviation of 3, 4, 5, 7, 10, 10, 10
Answer (Detailed Solution Below)
Elementary Statistics Question 12 Detailed Solution
Download Solution PDFGiven
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
The standard deviation of a data set is given as 34. What will be the variance of the data set?
Answer (Detailed Solution Below)
Elementary Statistics Question 13 Detailed Solution
Download Solution PDFGIVEN :
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 :
Variance of the set of data = 342 = 1156In a frequency distribution, the mid value of a class is 12 and its width is 6. The lower limit of the class is:
Answer (Detailed Solution Below)
Elementary Statistics Question 14 Detailed Solution
Download Solution PDFGiven:
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
Find the standard deviation of {7, 13, 15, 11, 4}
Answer (Detailed Solution Below)
Elementary Statistics Question 15 Detailed Solution
Download Solution PDFGiven:
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