MCQs on Statistics:
1. Who defined statistics as "the science of averages and their estimates"?
  a) Karl Pearson
  b) Sir Ronald A. Fisher
  c) A.L. Bowley & A.L. Boddington
  d) Francis Galton
  Answer: c) A.L. Bowley & A.L. Boddington
2. Which measure of central tendency is NOT commonly used to calculate the mean deviation?
  a) Mean
  b) Median
  c) Mode
  d) Range
  Answer: d) Range
3. What is the range of scores for Batsman A in the example given in the content?
  a) 117
  b) 14
  c) 0
  d) 53
  Answer: a) 117
4. Which of the following is NOT a measure of dispersion mentioned in the chapter?
  a) Range
  b) Quartile Deviation
  c) Mean Deviation
  d) Mean
  Answer: d) Mean
5. For a data set, the deviation from a central tendency (mean or median) is measured using which of the following?
  a) Range
  b) Standard deviation
  c) Mode
  d) Quartile deviation
  Answer: b) Standard deviation
6. What is the formula to calculate the mean of deviations about a central value ‘a’?
  a) $M.D.(a) = \frac{\text{Sum of deviations}}{\text{Number of observations}}$
  b) $M.D.(a) = \frac{\text{Sum of absolute values of deviations from } a}{\text{Number of observations}}$
  c) $M.D.(a) = \frac{\text{Sum of maximum and minimum values}}{2}$
  d) $M.D.(a) = \frac{\text{Sum of squares of deviations}}{\text{Number of observations}}$
  Answer: b) $M.D.(a) = \frac{\text{Sum of absolute values of deviations from } a}{\text{Number of observations}}$
7. What is the median for the data set: 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21?
  a) 9
  b) 10
  c) 12
  d) 7
  Answer: a) 9
8. What does the range of a data set indicate?
  a) Measure of central tendency
  b) Measure of variability between maximum and minimum values
  c) Measure of the mean deviation
  d) Measure of dispersion around the mean
  Answer: b) Measure of variability between maximum and minimum values
9. In grouped data, the median is found using which key component?
  a) Mid-point of the median class
  b) Frequency of the median class
  c) Lower boundary of the median class
  d) All of the above
  Answer: d) All of the above
10. Who is known as the Father of Modern Statistics?
  a) Karl Pearson
  b) Sir Ronald A. Fisher
  c) Francis Galton
  d) Captain John Graunt
  Answer: b) Sir Ronald A. Fisher
What does a measure of central tendency represent in a data set?
a) The spread of the data
b) The central value where data points cluster
c) The maximum value of the data set
d) The lowest value of the data set
Answer: b) The central value where data points cluster
Which measure of central tendency is most affected by extreme values?
a) Mean
b) Median
c) Mode
d) Range
Answer: a) Mean
In the calculation of mean deviation, why are absolute deviations used?
a) To simplify the calculation
b) To avoid negative values canceling positive ones
c) To emphasize larger deviations
d) None of the above
Answer: b) To avoid negative values canceling positive ones
If the number of observations is odd, how is the median calculated?
a) Mean of the two middle observations
b) The value of the middle observation after arranging in order
c) The mode of the dataset
d) The average of all observations
Answer: b) The value of the middle observation after arranging in order
Which of the following measures is used to describe the spread of data points?
a) Mean
b) Median
c) Mode
d) Dispersion
Answer: d) Dispersion

Range and Variability

What is the formula to calculate the range of a dataset?
a) Maximum value + Minimum value
b) Maximum value - Minimum value
c) Sum of values ÷ Total observations
d) Maximum value × Minimum value
Answer: b) Maximum value - Minimum value
If the range of a dataset is small, it indicates that the data is:
a) More dispersed
b) Clustered closely
c) Normally distributed
d) Symmetrical
Answer: b) Clustered closely
The range of the scores of Batsman B (46 to 60) is:
a) 14
b) 60
c) 46
d) 53
Answer: a) 14
What additional factor is needed to fully interpret the variability of data beyond central tendency?
a) Mode
b) Dispersion
c) Median
d) Sum of observations
Answer: b) Dispersion
What is the limitation of using range as a measure of dispersion?
a) It ignores variability within the dataset
b) It is hard to calculate
c) It only considers central values
d) It does not work for grouped data
Answer: a) It ignores variability within the dataset

Mean Deviation

What is the mean deviation of a dataset where the absolute deviations from the mean total 30 and there are 6 observations?
a) 30
b) 6
c) 5
d) 3
Answer: c) 5
Which measure of dispersion depends on the deviations of data from a central value?
a) Range
b) Mean deviation
c) Mode
d) Quartile deviation
Answer: b) Mean deviation
For grouped data, the mean deviation about the mean is calculated using:
a) Class midpoints and frequencies
b) Individual data points
c) Sum of squared deviations
d) Upper and lower boundaries of each class
Answer: a) Class midpoints and frequencies
The sum of deviations from the mean in a dataset is always:
a) Equal to the range
b) Zero
c) Positive
d) Negative
Answer: b) Zero
The absolute mean deviation from the median of a dataset indicates:
a) Central tendency
b) Variability around the median
c) Highest and lowest values
d) Mean value of deviations
Answer: b) Variability around the median

Standard Deviation

What is the standard deviation?
a) The square root of the variance
b) The mean of the dataset
c) The absolute deviation from the mean
d) The range divided by the total observations
Answer: a) The square root of the variance
What does a high standard deviation indicate?
a) The data points are closely clustered around the mean
b) The data points are more spread out from the mean
c) There is no variability in the data
d) The range is small
Answer: b) The data points are more spread out from the mean
If the standard deviation of a dataset is 5, what is the variance?
a) 25
b) 5
c) 10
d) 15
Answer: a) 25
How is the variance of a dataset calculated?
a) Mean of deviations from the median
b) Sum of squared deviations from the mean divided by the number of observations
c) Absolute deviations divided by the range
d) Sum of deviations from the maximum value
Answer: b) Sum of squared deviations from the mean divided by the number of observations
If all data values are the same in a dataset, the standard deviation will be:
a) 1
b) 0
c) Equal to the mean
d) Undefined
Answer: b) 0

Grouped Data

In a frequency distribution, the midpoint of a class is calculated as:
a) The difference between the upper and lower boundaries
b) The average of the upper and lower boundaries
c) The product of frequency and range
d) None of the above
Answer: b) The average of the upper and lower boundaries
What is the formula for mean deviation about the mean for grouped data?
a) $\text{M.D.} = \frac{\Sigma f \cdot |x - \bar{x}|}{N}$
b) $\text{M.D.} = \frac{\Sigma |x - \bar{x}|}{N}$
c) $\text{M.D.} = \frac{\Sigma f \cdot |x - \text{median}|}{N}$
d) $\text{M.D.} = \frac{\Sigma |x - \text{mean}|}{N}$
Answer: a) $\text{M.D.} = \frac{\Sigma f \cdot |x - \bar{x}|}{N}$
For continuous frequency distribution, the standard deviation is calculated using:
a) Class limits
b) Cumulative frequencies
c) Class midpoints and frequencies
d) Range
Answer: c) Class midpoints and frequencies
If each observation in a dataset is multiplied by a constant $k$ , the variance is:
a) $k \times \text{original variance}$
b) $k^2 \times \text{original variance}$
c) The same as the original variance
d) $\frac{\text{original variance}}{k}$
Answer: b) $k^2 \times \text{original variance}$
In the shortcut method, deviations are taken from:
a) Class midpoints
b) Assumed mean
c) Mean deviation
d) Range
Answer: b) Assumed mean

Historical Development of Statistics

Who is considered the "Father of Modern Statistics"?
a) Karl Pearson
b) Sir Ronald A. Fisher
c) Jacob Bernoulli
d) Francis Galton
Answer: b) Sir Ronald A. Fisher
The first known census was conducted in which country?
a) India
b) Egypt
c) England
d) China
Answer: b) Egypt
Kautilya’s Arthashastra mentions a system for collecting which type of data?
a) Weather statistics
b) Birth and death statistics
c) Agricultural statistics
d) Trade statistics
Answer: b) Birth and death statistics
The theory of probability began development in which century?
a) 16th century
b) 17th century
c) 18th century
d) 19th century
Answer: b) 17th century
Who introduced the Chi-square test?
a) Karl Pearson
b) Francis Galton
c) Ronald A. Fisher
d) Abul Fazl
Answer: a) Karl Pearson

Measures of Dispersion

What is the primary purpose of measuring dispersion in a dataset?
a) To find the average value of data
b) To analyze the spread or variability of data
c) To identify the median
d) To determine the maximum value
Answer: b) To analyze the spread or variability of data
Which of the following is NOT a measure of dispersion?
a) Range
b) Mean deviation
c) Variance
d) Arithmetic mean
Answer: d) Arithmetic mean
What is the range if the maximum value is 75 and the minimum value is 15?
a) 60
b) 90
c) 15
d) 75
Answer: a) 60
What is the main limitation of using the range as a measure of dispersion?
a) It is difficult to calculate
b) It considers only two values in the dataset
c) It cannot be used for grouped data
d) It depends on the mean
Answer: b) It considers only two values in the dataset
If the range of Dataset A is greater than Dataset B, it implies that Dataset A is:
a) More consistent
b) More scattered
c) Less variable
d) Symmetrical
Answer: b) More scattered

Mean Deviation and Variance

Which of the following is used to calculate the mean deviation about the mean?
a) Absolute deviations from the mean
b) Squared deviations from the mean
c) Total sum of observations
d) Difference between maximum and minimum values
Answer: a) Absolute deviations from the mean
What does the variance measure in a dataset?
a) Central tendency
b) Average of squared deviations from the mean
c) Difference between mean and median
d) Relationship between mean and mode
Answer: b) Average of squared deviations from the mean
If the variance of a dataset is zero, it means:
a) All observations are the same
b) The dataset has no central tendency
c) The dataset is highly variable
d) The standard deviation is infinite
Answer: a) All observations are the same
How does multiplying all observations in a dataset by a constant $k$ affect the variance?
a) Variance remains unchanged
b) Variance becomes $k^2$ times the original variance
c) Variance is halved
d) Variance becomes $k$ times the original variance
Answer: b) Variance becomes $k^2$ times the original variance
The mean of squared deviations from the mean of a dataset is called:
a) Mean deviation
b) Range
c) Variance
d) Standard deviation
Answer: c) Variance

Standard Deviation and Grouped Data

Standard deviation is always:
a) Negative
b) Zero or positive
c) Greater than variance
d) Equal to the mean deviation
Answer: b) Zero or positive
Which step is necessary to calculate the standard deviation for grouped data?
a) Finding midpoints of classes
b) Calculating cumulative frequencies
c) Determining the range
d) Summing deviations without squaring
Answer: a) Finding midpoints of classes
In a frequency distribution, the total frequency is denoted by:
a) $f$
b) $N$
c) $\sigma$
d) $x$
Answer: b) $N$
In the shortcut method for variance, deviations are taken from:
a) Arithmetic mean
b) Assumed mean
c) Median
d) Mode
Answer: b) Assumed mean
In a continuous frequency distribution, the median class is identified using:
a) The midpoint of each class
b) The cumulative frequency table
c) The total sum of observations
d) The range of the dataset
Answer: b) The cumulative frequency table

Historical Context of Statistics

Which book by Jacob Bernoulli discussed the Law of Large Numbers?
a) Arthashastra
b) Ars Conjectandi
c) Ain-i-Akbari
d) Chi-Square Theory
Answer: b) Ars Conjectandi
Francis Galton is known for pioneering statistical methods in which field?
a) Genetics
b) Biometry
c) Astronomy
d) Agriculture
Answer: b) Biometry
Captain John Graunt is recognized for his contributions to:
a) Biometry
b) Vital statistics
c) Agricultural surveys
d) Probability theory
Answer: b) Vital statistics
During Akbar’s reign, surveys on administration were documented in:
a) Ars Conjectandi
b) Arthashastra
c) Ain-i-Akbari
d) Chi-Square Laboratory
Answer: c) Ain-i-Akbari
Who introduced the Chi-Square test in statistics?
a) Karl Pearson
b) Sir Ronald Fisher
c) Jacob Bernoulli
d) Francis Galton
Answer: a) Karl Pearson

Applications of Statistics

In statistics, the term "dispersion" refers to:
a) Central tendency of data
b) Spread or variability of data points
c) The most frequently occurring value
d) The arithmetic mean of the dataset
Answer: b) Spread or variability of data points
When is the range an ideal measure of dispersion?
a) When comparing two datasets
b) When the data is symmetric
c) For small datasets with no extreme values
d) Always
Answer: c) For small datasets with no extreme values
Mean deviation can be calculated from which measures of central tendency?
a) Mean and median
b) Mean and range
c) Median and mode
d) Mode and range
Answer: a) Mean and median
If the cumulative frequency is 30, the median class lies at:
a) $N/4$
b) $N/2$
c) $N-1$
d) $3N/4$
Answer: b) $N/2$
Which measure of dispersion is commonly used in modern statistical applications?
a) Range
b) Quartile deviation
c) Standard deviation
d) Mode
Answer: c) Standard deviation

Problem Solving in Statistics

The median of the dataset: 4, 8, 10, 15, 20, 22, 30 is:
a) 15
b) 10
c) 20
d) 18
Answer: a) 15
What happens to the mean if 5 is added to every observation in a dataset?
a) Mean increases by 5
b) Mean decreases by 5
c) Mean doubles
d) Mean remains unchanged
Answer: a) Mean increases by 5
If the standard deviation of a dataset is $\sigma$ , the variance is given by:
a) $\sigma/2$
b) $\sigma^2$
c) $2\sigma$
d) $1/\sigma$
Answer: b) $\sigma^2$
In grouped data, frequencies are multiplied by which value to calculate the mean?
a) Cumulative frequencies
b) Midpoints of the class
c) Range of the class
d) Class intervals
Answer: b) Midpoints of the class
Which step is necessary to calculate the variance of a dataset?
a) Squaring deviations from the mean
b) Finding the absolute deviations from the mode
c) Multiplying deviations by cumulative frequencies
d) Taking the square root of deviations from the mean
Answer: a) Squaring deviations from the mean

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Questions & Answers HS (11th & 12th)

Friday, November 22, 2024

MCQ Mathematics Chapter 13: STATISTICS, HS 1st year

MCQs on Statistics: