A dataset has a mean of 50 and a variance of 25. If each data point is divided by 5, what is the new variance?

Correct Option: A. Solution: The variance of a dataset when each data point is divided by a constant $ c $ is $ \frac{\sigma^2}{c^2} $. Here, $ \sigma^2 = 25 $ and $ c = 5 $. Thus, the new variance is $ \frac{25}{5^2} = 1 $.

Statistics

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Summary

Chapter Summary: Statistics

Key Concepts

Statistics: Science of averages and their estimates.
Measures of Central Tendency: Mean, Median, Mode.
Measures of Dispersion: Range, Quartile deviation, Mean deviation, Variance, Standard deviation.

Important Formulas

Mean Deviation (M.D.):
- For ungrouped data: M.D. = $\frac{\Sigma |x_i - M|}{n}$
- For grouped data: M.D. = $\frac{N}{N}$
Variance (σ²):
- For ungrouped data: $\sigma^2 = \frac{\Sigma (x_i - \mu)^2}{n}$
- For grouped data: $\sigma^2 = \frac{\Sigma f_i (x_i - A)^2}{N}$
Standard Deviation (σ): $\sigma = \sqrt{\sigma^2}$

Examples

Example of Mean Deviation Calculation:
- Data: 6, 7, 10, 12, 13, 4, 8, 12
- Mean (x) = 9
- Deviations: -3, -2, 1, 3, 4, -5, -1, 3
- Mean Deviation = 2.75

Common Pitfalls

Ignoring Class Intervals: Ensure to convert data into continuous frequency distribution when necessary.
Miscalculating Deviations: Always check calculations of deviations from mean or median.

Tips for Exam Preparation

Practice calculating mean, variance, and standard deviation using both direct and shortcut methods.
Familiarize yourself with frequency distribution tables and how to interpret them.

Learning Objectives

Understand the concept of mean, median, and mode as measures of central tendency.
Calculate mean deviation about the mean and median for given data.
Apply the shortcut method for calculating variance and standard deviation.
Interpret frequency distribution tables and extract relevant statistical measures.
Recognize the historical development of statistics and its applications in various fields.
Analyze data sets to determine measures of dispersion such as range, quartile deviation, variance, and standard deviation.

Detailed Notes

Chapter 13: Statistics

13.1 Introduction

Statistics deals with data collected for specific purposes.
It involves analyzing and interpreting data to make decisions.
Key measures of central tendency:
- Mean (Arithmetic Mean)
- Median
- Mode

13.2 Measures of Dispersion

Measures of dispersion include:
- Range
- Quartile Deviation
- Mean Deviation
- Variance
- Standard Deviation

13.2.1 Mean Deviation

Mean Deviation (M.D.) for ungrouped data:
- M.D. = $\frac{\Sigma |x_i - M|}{n}$

13.2.2 Variance and Standard Deviation

Variance for ungrouped data:
- Variance = $\frac{\Sigma (x_i - \bar{x})^2}{n}$
Standard Deviation = $\sqrt{Variance}$

13.3 Frequency Distribution

Example: Mean Deviation Calculation

Given data: Marks obtained and number of students.

Marks obtained	Number of students	Mid-points	fᵢ	xᵢ	fᵢdᵢ	xᵢ - x	fᵢ	xᵢ - x
10-20	2	15	2	15	30	-3	60	--
20-30	3	25	3	25	75	-2	60	--
30-40	8	35	8	35	280	-1	80	--
40-50	14	45	14	45	630	0	0	--
50-60	8	55	8	55	440	1	80	--
60-70	3	65	3	65	195	2	60	--
70-80	2	75	2	75	150	3	60	--
	40				1800		400	--

13.4 Limitations of Mean Deviation

In series with high variability, the median may not be representative.
Mean deviation may yield unsatisfactory results.

13.5 Historical Note

The term 'Statistics' is derived from the Latin word 'status'.
Significant historical figures include:
- Captain John Graunt: Father of vital statistics.
- Jacob Bernoulli: Stated the Law of Large Numbers.

Exercises

Find the mean deviation about the mean for the following data:
- 6, 7, 10, 12, 13, 4, 8, 12
Calculate the mean and variance for the frequency distributions provided.

Example: Variance Calculation

Given data:

Classes	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Frequency	3	7	12	15	8	3	2

Calculate mean, variance, and standard deviation using the shortcut method.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Ignoring the Median: Students often forget to use the median when calculating mean deviation about the median, leading to incorrect results.
Calculation Errors: Mistakes in arithmetic when calculating deviations can skew results significantly.
Misunderstanding Class Intervals: Confusion can arise when interpreting class intervals, especially in frequency distributions. Ensure to convert data into continuous intervals if required.
Not Using Absolute Values: When calculating mean deviation, failing to take absolute values of deviations can result in negative sums, which are not meaningful in this context.

Tips for Success

Follow Step-by-Step Procedures: Always break down calculations into clear steps to avoid errors. For example, when finding mean deviation, calculate the mean first, then the deviations, and finally the mean deviation.
Double-Check Your Work: After calculations, review each step to ensure accuracy, especially in summing deviations and frequencies.
Use Tables for Clarity: Organize data in tables to visualize calculations better, especially when dealing with large datasets or frequency distributions.
Practice with Examples: Work through multiple examples to become familiar with the process of calculating mean deviation and variance. This will help reinforce the concepts and reduce mistakes during exams.

Practice & Assessment

Multiple Choice Questions

2.5

2.75

3.0

3.25

Correct Answer: B

Solution:

The mean deviation about the mean is calculated as 2.75 for the given data set.

Correct Answer: B

Solution:

When each observation is multiplied by 3, the new mean will also be multiplied by 3: 10 * 3 = 30.

Francis Galton

Karl Pearson

Sir Ronald A. Fisher

Jacob Bernoulli

Correct Answer: C

Solution:

Sir Ronald A. Fisher is known as the Father of modern statistics due to his contributions to various fields such as Genetics, Biometry, Education, and Agriculture.

It is calculated using the absolute deviations from the median.

It is always greater than the standard deviation.

It can be subjected to further algebraic treatment.

It is a measure of central tendency.

Correct Answer: A

Solution:

Mean deviation is calculated using the absolute deviations from the mean or median, not the standard deviation. It is not a measure of central tendency and cannot be subjected to further algebraic treatment.

Correct Answer: A

Solution:

When each observation in a dataset is increased by a constant, the variance remains unchanged. Therefore, the new variance is 16.

Correct Answer: D

Solution:

The range is calculated as the difference between the maximum and minimum values: 12 - 2 = 10.

Correct Answer: D

Solution:

Variance is the square of the standard deviation: 4^2 = 16.

Captain John Graunt

Jacob Bernoulli

Karl Pearson

Sir Ronald A. Fisher

Correct Answer: A

Solution:

Captain John Graunt of London is known as the father of vital statistics due to his studies on statistics of births and deaths.

It is always greater than the standard deviation.

It is calculated using absolute deviations from the mean.

It can be negative.

It is the same as variance.

Correct Answer: B

Solution:

Mean deviation is calculated using the absolute values of deviations from the mean.

Range

Mean

Standard Deviation

Variance

Correct Answer: B

Solution:

Mean is a measure of central tendency, not a measure of dispersion. Measures of dispersion include Range, Standard Deviation, and Variance.

39.8

39.9

40.0

40.1

Correct Answer: A

Solution:

The incorrect observation contributes 45 to the sum of observations. Replacing it with 35 decreases the sum by 10. New mean = (Original sum - 10)/50 = (40 × 50 - 10)/50 = 39.8.

Correct Answer: B

Solution:

The median is calculated using the formula:

\text{Median} = l + \frac{\left(\frac{N}{2} - C\right)}{f} \times h

where

l = 20

C = 13

f = 15

h = 10

. Thus, Median =

20 + \frac{(25 - 13)}{15} \times 10 = 28

Median class

Mean class

Mode class

Range class

Correct Answer: C

Solution:

The class interval with the highest frequency in a continuous frequency distribution is known as the mode class.

Correct Answer: B

Solution:

The mean is calculated as the sum of all observations divided by the number of observations. Therefore, the number of observations is

\frac{800}{40} = 20

Biometry

Genetics

Agriculture

Education

Correct Answer: A

Solution:

Francis Galton is known for pioneering the use of statistical methods in the field of Biometry.

10-20

20-30

30-40

40-50

Correct Answer: B

Solution:

The median class is determined by the cumulative frequency, which places it in the 20-30 interval.

Correct Answer: D

Solution:

Variance is the square of the standard deviation. Therefore, variance = 5^2 = 25.

Correct Answer: B

Solution:

The number of observations is calculated by dividing the sum of all observations by the mean: 200 / 20 = 10.

250

Correct Answer: A

Solution:

The range is the difference between the maximum and minimum values. If each observation is divided by 5, the range also gets divided by 5. Therefore, the new range is

50 \div 5 = 10

The average of a data set

The spread of data points around a central value

The sum of all data points

The difference between the highest and lowest data points

Correct Answer: B

Solution:

Dispersion refers to how much the data points are spread out around a central value.

Correct Answer: A

Solution:

The standard deviation is the square root of the variance:

\sqrt{4} = 2

Range

Mean deviation about the mean

Mode

Quartile deviation

Correct Answer: B

Solution:

The mid-point of each class interval is used to calculate the mean deviation about the mean in a continuous frequency distribution.

Correct Answer: C

Solution:

When each observation in a dataset is multiplied by a constant, the standard deviation is also multiplied by that constant. Therefore, the new standard deviation will be

3 \times 15 = 45

144

Correct Answer: C

Solution:

When each data point is multiplied by a constant, the variance is multiplied by the square of that constant. Therefore, the new variance is

16 \times 3^2 = 144

Correct Answer: A

Solution:

If each observation is divided by a constant, the new variance is the old variance divided by the square of that constant. Thus, the new variance is 16 / 4 = 4.

2.75

3.25

3.00

2.50

Correct Answer: A

Solution:

The mean deviation about the mean is calculated as the average of the absolute deviations from the mean. For this data set, it is calculated as

\frac{22}{8} = 2.75

Correct Answer: C

Solution:

When each observation is multiplied by 3, the new mean is also multiplied by 3, resulting in 24.

117

112

Correct Answer: A

Solution:

The range is calculated as the maximum value minus the minimum value: 117 - 0 = 117.

Correct Answer: A

Solution:

The standard deviation is the square root of the variance. Thus, if the variance is 9, the standard deviation is

\sqrt{9} = 3

The difference between the maximum and minimum values

The average of all values

The middle value when data is arranged in order

The most frequently occurring value

Correct Answer: A

Solution:

Range is calculated as the difference between the maximum and minimum values of a data set.

Correct Answer: A

Solution:

When each observation in a data set is decreased by a constant, the mean is also decreased by that constant. Therefore, the new mean is

50 - 5 = 45

Mean

Median

Mode

Range

Correct Answer: B

Solution:

The median is the most appropriate measure of central tendency to use when a data set has extreme values, as it is not affected by outliers.

Mean

Median

Range

Mode

Correct Answer: C

Solution:

Range is a measure of dispersion as it indicates the spread of data.

Mean deviation

Standard deviation

Range

Interquartile range

Correct Answer: D

Solution:

The interquartile range is a measure of dispersion that is less affected by skewed data and outliers compared to other measures like range or standard deviation.

Correct Answer: A

Solution:

The standard deviation is the square root of the variance. Therefore, the standard deviation is

\sqrt{16} = 4

Francis Galton

Karl Pearson

Sir Ronald A. Fisher

A.L. Bowley

Correct Answer: B

Solution:

Karl Pearson contributed to the development of statistical studies with his discovery of the Chi square test.

Correct Answer: A

Solution:

Standard deviation is the square root of variance. Therefore, the standard deviation is √9 = 3.

Correct Answer: B

Solution:

When each observation is increased by a constant, the mean also increases by that constant. Therefore, the new mean is 20 + 5 = 25.

Correct Answer: A

Solution:

The range is the difference between the maximum and minimum values. Increasing each observation by 5 does not change the range, so the new range remains 50.

Correct Answer: A

Solution:

The variance of a dataset when each data point is divided by a constant

c

\frac{\sigma^2}{c^2}

. Here,

\sigma^2 = 25

and

c = 5

. Thus, the new variance is

\frac{25}{5^2} = 1

Correct Answer: C

Solution:

When each observation in a data set is multiplied by a constant, the mean is also multiplied by that constant. Thus, the new mean is 20 \times 3 = 60.

Rigveda

Mahabharata

Arthashastra

Ramayana

Correct Answer: C

Solution:

The Arthashastra, an ancient Indian text attributed to Kautilya, mentions the collection of administrative statistics, particularly during the regime of Chandra Gupta Maurya.

Correct Answer: B

Solution:

When each observation is increased by a constant, the mean also increases by the same constant. Therefore, the new mean is 15 + 5 = 20.

Species A

Species B

Both have the same variability

Cannot be determined

Correct Answer: B

Solution:

Species B has a higher standard deviation (15 cm) compared to Species A (10 cm), indicating greater variability in height.

Correct Answer: A

Solution:

Adding a constant to each observation does not change the standard deviation. Therefore, the new standard deviation remains 5.

The median is exactly 25

The median is greater than 25

The median is less than 25

The median is exactly 30

Correct Answer: A

Solution:

The median class is the class interval where the cumulative frequency reaches or exceeds half of the total frequency. Since the median class is 20-30, the median is calculated using the formula for the median of a grouped frequency distribution, which typically results in a value around the midpoint of the class interval, in this case, 25.

T-test

Chi square test

ANOVA

Z-test

Correct Answer: B

Solution:

Karl Pearson is credited with the discovery of the Chi square test.

Correct Answer: B

Solution:

When each observation in a dataset is multiplied by a constant

a

, the new standard deviation becomes

a \times \sigma

. Thus, the new standard deviation is

3 \times 4 = 12

Genetics

Biometry

Astronomy

Agriculture

Correct Answer: C

Solution:

Sir Ronald A. Fisher applied statistical methods to fields such as Genetics, Biometry, and Agriculture, but not to Astronomy.

Francis Galton

Karl Pearson

Sir Ronald A. Fisher

John Graunt

Correct Answer: B

Solution:

Karl Pearson is credited with the discovery of the Chi square test.

Mean deviation

Range

Standard deviation

Quartile deviation

Correct Answer: C

Solution:

Standard deviation is the measure of dispersion calculated as the square root of the variance.

Correct Answer: A

Solution:

The number of observations is the sum of all observations divided by the mean: 300 / 30 = 10.

Range

Mean deviation

Standard deviation

Median

Correct Answer: D

Solution:

Median is a measure of central tendency, not a measure of dispersion.

Correct Answer: C

Solution:

When each observation in a dataset is multiplied by a constant, the mean deviation is also multiplied by that constant. Therefore, the new mean deviation is 5 × 3 = 15.

117

Correct Answer: B

Solution:

The range is calculated as the maximum value minus the minimum value. For Batsman A, it is 117 - 0 = 117.

Correct Answer: B

Solution:

The median is the middle value of an ordered data set. Here, the median is 7.

Positive

Negative

Zero

Equal to the variance

Correct Answer: C

Solution:

By definition, the sum of the deviations from the mean for any data set is always zero because the mean is a measure of central tendency.

144

Correct Answer: C

Solution:

When each data point is multiplied by a constant, the variance is multiplied by the square of that constant. Therefore, the new variance is

4^2 \times 9 = 144

150

120

180

200

Correct Answer: A

Solution:

The sum of the data set is calculated by multiplying the mean by the number of observations: 30 * 5 = 150.

Correct Answer: A

Solution:

The range is calculated as the difference between the maximum and minimum values: 55 - 5 = 50.

100

400

625

Correct Answer: C

Solution:

When each observation is multiplied by a constant, the variance is multiplied by the square of that constant. Therefore, the new variance is

25 \times 4^2 = 400

The data is highly dispersed.

The data is closely clustered around the mean.

The data has no dispersion.

The data is normally distributed.

Correct Answer: A

Solution:

A large sum of squares of deviations indicates a higher degree of dispersion from the mean.

32.5

Correct Answer: A

Solution:

The median of a class interval is the midpoint of the interval. For the class 30-40, the midpoint is

\frac{30 + 40}{2} = 35

Correct Answer: A

Solution:

The standard deviation is a measure of dispersion and is not affected by adding a constant to all observations. Therefore, the new standard deviation remains 5.

Dataset X

Dataset Y

Both have the same dispersion

Cannot be determined

Correct Answer: B

Solution:

The range of a dataset is a measure of dispersion. Since Dataset Y has a larger range (60) compared to Dataset X (40), it has more dispersion.

Correct Answer: A

Solution:

The variance remains unchanged when a constant is added to each data point. Therefore, the new variance is 25.

Range

Standard Deviation

Variance

Mean Deviation

Correct Answer: A

Solution:

The range is defined as the difference between the maximum and minimum values of a dataset.

Correct Answer: B

Solution:

The number of observations is calculated by dividing the sum of all observations by the mean: 500/50 = 10.

The middle value

The average of the two middle values

The most frequent value

The difference between the maximum and minimum values

Correct Answer: B

Solution:

For an even number of observations, the median is the average of the two middle values.

The data points are closely clustered around the mean.

The data points are widely spread out from the mean.

The mean deviation is not a reliable measure of dispersion.

The dataset has a high mean value.

Correct Answer: A

Solution:

A lower mean deviation indicates that the data points are closely clustered around the mean, implying less variability.

Mean

Median

Mode

Range

Correct Answer: A

Solution:

The mean is the most affected by extreme values because it is calculated by summing all values and dividing by the number of values, whereas the median and mode are measures of central tendency that are less influenced by outliers.

2.5

Correct Answer: A

Solution:

When each data point is divided by a constant, the standard deviation is also divided by that constant. Thus, the new standard deviation is 5 / 2 = 2.5.

17.5

22.5

27.5

32.5

Correct Answer: B

Solution:

The original mean is calculated as

\frac{5 + 7 + 9 + 12 + 14 + 15 + 18 + 22 + 24 + 30}{10} = 15.6

. If each observation is increased by 5, the new mean will be

15.6 + 5 = 20.6

Correct Answer: A

Solution:

The range is calculated as the difference between the maximum and minimum values: 31 - 8 = 23.

Increase

Decrease

Remain the same

Cannot be determined

Correct Answer: A

Solution:

Replacing an observation with a value further from the mean increases the spread of the data, thus increasing the standard deviation.

Mean

Variance

Standard Deviation

Range

Correct Answer: A

Solution:

Mean is a measure of central tendency, while variance, standard deviation, and range are measures of dispersion.

Range is a measure of central tendency.

Range is the difference between the maximum and minimum values.

Range is always equal to the mean.

Range is the sum of all observations.

Correct Answer: B

Solution:

Range is defined as the difference between the maximum and minimum values of a dataset.

Correct Answer: A

Solution:

The range is calculated as the difference between the maximum and minimum values. Here, Range = 50 - 10 = 40.

Correct Answer: B

Solution:

The standard deviation is the square root of the variance. Therefore, if the variance is 16, the standard deviation is

\sqrt{16} = 4

True or False

Correct Answer: True

Solution:

Karl Pearson contributed to the development of statistical studies with his discovery of the Chi square test.

Correct Answer: False

Solution:

Captain John Graunt is known as the father of vital statistics due to his studies on statistics of births and deaths, not modern statistics.

Correct Answer: True

Solution:

Mean deviation is calculated by taking the absolute values of deviations from the mean to avoid cancellation of positive and negative deviations.

Correct Answer: True

Solution:

The range is calculated as the difference between the maximum and minimum values of a data set.

Correct Answer: True

Solution:

The range of a data set is the difference between the maximum and minimum values, providing a measure of variability.

Correct Answer: False

Solution:

Sir Ronald A. Fisher is known as the Father of modern statistics, not Francis Galton.

Correct Answer: True

Solution:

Karl Pearson contributed significantly to statistical studies, including the discovery of the Chi square test.

Correct Answer: False

Solution:

The first known census was conducted in Egypt around 3050 B.C., not in India.

Correct Answer: True

Solution:

Karl Pearson is credited with the development of the Chi square test, which is a significant contribution to statistical studies.

Correct Answer: False

Solution:

The mean deviation about the median is not always less than the mean deviation about the mean; it depends on the data distribution.

Correct Answer: False

Solution:

The mean deviation is a measure of dispersion, not central tendency.

Correct Answer: True

Solution:

Sir Ronald A. Fisher made significant contributions to statistics and is recognized as the Father of modern statistics.

Correct Answer: True

Solution:

The range is defined as the difference between the maximum and minimum values of a data set.

Correct Answer: False

Solution:

The mean deviation about the mean is not zero; it is calculated using the absolute values of deviations.

Correct Answer: True

Solution:

The excerpts mention that the theoretical development of statistics came during the mid-seventeenth century and continued with the introduction of probability.

Correct Answer: False

Solution:

The range is a measure of dispersion, not central tendency. It indicates the spread of the data.

Correct Answer: True

Solution:

The range is defined as the difference between the maximum and minimum values in a data set.

Correct Answer: False

Solution:

The range is a measure of dispersion, not a measure of central tendency.

Correct Answer: False

Solution:

Standard deviation is a measure of dispersion, not central tendency. It indicates how spread out the values in a data set are.

Correct Answer: False

Solution:

The mean deviation is a measure of dispersion, not central tendency.

Correct Answer: False

Solution:

The excerpt explains that 'statistics' is derived from the Latin word 'status', not the Greek word 'statikos'.

Correct Answer: False

Solution:

The median is the middle value of a data set when arranged in order, but if the number of observations is even, the median is the average of the two middle values, which may not be an actual data point.

Correct Answer: True

Solution:

The word 'statistics' comes from the Latin word 'status', which means a political state.

Correct Answer: True

Solution:

The sum of deviations from the mean is always zero because the mean is the balance point of the data.

Correct Answer: True

Solution:

Jacob Bernoulli stated the Law of Large Numbers in his book 'Ars Conjectandi', published in 1713.

Correct Answer: False

Solution:

The range gives a rough idea of variability but does not provide complete information about the dispersion of data from a measure of central tendency.

Correct Answer: True

Solution:

By definition, the sum of the deviations from the mean is zero because the mean is the balance point of the data.

Correct Answer: True

Solution:

According to the excerpts, Karl Pearson contributed to the development of statistical studies with his discovery of the Chi square test and the foundation of the statistical laboratory in England in 1911.

Correct Answer: True

Solution:

Standard deviation is a measure of dispersion that indicates how much data points deviate from the mean.

Correct Answer: True

Solution:

The excerpt states that the theoretical development of statistics started during the mid-seventeenth century and continued with the introduction of the theory of games and chance.

Correct Answer: False

Solution:

Mean deviation is calculated on the basis of absolute values of the deviations and therefore cannot be subjected to further algebraic treatment.

Correct Answer: True

Solution:

Sir Ronald A. Fisher is often referred to as the Father of modern statistics due to his contributions to various fields.

Correct Answer: False

Solution:

Mean deviation is calculated on the basis of absolute values of the deviations and cannot be subjected to further algebraic treatment.

Correct Answer: True

Solution:

Francis Galton, an Englishman, is credited with pioneering the use of statistical methods in the field of Biometry.

Correct Answer: False

Solution:

Measures of central tendency such as mean, median, and mode do not provide complete information about data variability. Variability needs to be studied separately.

Correct Answer: True

Solution:

The sum of squared deviations from the mean is non-negative and only zero if all observations are equal to the mean.

Correct Answer: False

Solution:

The Chi square test was discovered by Karl Pearson, not Sir Ronald A. Fisher.

Correct Answer: False

Solution:

The median and mean are different measures of central tendency and are not always equal. They can be the same in some cases, but not always.

Correct Answer: True

Solution:

The range, calculated as the difference between the maximum and minimum values, provides a rough measure of data variability.

Correct Answer: True

Solution:

Captain John Graunt is recognized as the father of vital statistics due to his pioneering work on the statistics of births and deaths.

Correct Answer: True

Solution:

Karl Pearson is credited with the development of the Chi square test.

Correct Answer: True

Solution:

Historical records suggest that the first known census was conducted in Egypt around 3050 B.C.

Correct Answer: True

Solution:

Captain John Graunt's studies on the statistics of births and deaths earned him the title of the father of vital statistics.

Correct Answer: False

Solution:

The standard deviation is a measure of dispersion, not central tendency.

Correct Answer: True

Solution:

Francis Galton is recognized for pioneering the use of statistical methods in Biometry.

Correct Answer: False

Solution:

The mean deviation about the median can be unreliable when the degree of variability is high, as it may not represent the central tendency well.

Correct Answer: False

Solution:

The mean deviation about the mean is not zero; it is a measure of dispersion calculated using absolute deviations.

Correct Answer: True

Solution:

The theoretical development of statistics started during the mid-seventeenth century with the introduction of theories related to games and chance, such as probability.

Correct Answer: True

Solution:

The sum of the absolute deviations from the mean is generally more than the sum of the deviations from the median.

Correct Answer: False

Solution:

The mean deviation about the mean is not very scientific as it is calculated on the basis of absolute values and cannot be subjected to further algebraic treatment.

Correct Answer: False

Solution:

The mean deviation about the mean is not always reliable, especially when the degree of variability is high, as it cannot be subjected to further algebraic treatment.

Correct Answer: True

Solution:

The word 'statistics' is derived from the Latin word 'status', which means a political state, indicating that the use of statistics is as old as human civilization.

Correct Answer: False

Solution:

Sir Ronald A. Fisher is known as the Father of modern statistics, not Karl Pearson.

Correct Answer: False

Solution:

The excerpts identify Captain John Graunt as the father of vital statistics, not modern statistics. Sir Ronald A. Fisher is often referred to as the father of modern statistics.

Correct Answer: False

Solution:

The first known census was held in Egypt in 3050 B.C., not in India during Chandra Gupta Maurya's regime.

Correct Answer: True

Solution:

The word 'statistics' originates from the Latin word 'status', which means a political state.

Correct Answer: True

Solution:

Range is one of the measures of dispersion, calculated as the difference between the maximum and minimum values of the data set.

Correct Answer: True

Solution:

The excerpt mentions that perhaps the first census was held in Egypt around 3050 B.C.

Statistics

AI Learning Assistant

Summary

Chapter Summary: Statistics

Key Concepts

Important Formulas

Examples

Common Pitfalls

Tips for Exam Preparation

Learning Objectives

Learning Objectives

Detailed Notes

Chapter 13: Statistics

13.1 Introduction

13.2 Measures of Dispersion

13.2.1 Mean Deviation

13.2.2 Variance and Standard Deviation

13.3 Frequency Distribution

Example: Mean Deviation Calculation

13.4 Limitations of Mean Deviation

13.5 Historical Note

Exercises

Example: Variance Calculation

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.What is the mean deviation about the mean for the data set: 6, 7, 10, 12, 13, 4, 8, 12?

Correct Answer: B

Q2.If the mean of a data set is 10 and each observation is multiplied by 3, what will be the new mean?

Correct Answer: B

Q3.Who is known as the Father of modern statistics?

Correct Answer: C

Q4.Which of the following statements is true about the mean deviation?

Correct Answer: A

Q5.Consider a dataset with a mean of 50 and a variance of 16. If each observation in the dataset is increased by 5, what will be the new variance?

Correct Answer: A

Q6.What is the range of the data set: 6, 7, 10, 12, 13, 4, 8, 12?

Correct Answer: D

Q7.If the standard deviation of a data set is 4, what is the variance?

Correct Answer: D

Q8.Which historical figure is known as the father of vital statistics due to his studies on statistics of births and deaths?

Correct Answer: A

Q9.Which of the following statements is true about the mean deviation?

Correct Answer: B

Q10.Which of the following is NOT a measure of dispersion?

Correct Answer: B

Q11.The mean and standard deviation of a group of 50 observations are found to be 40 and 5, respectively. If an incorrect observation of 45 is replaced by 35, what will be the new mean?

Correct Answer: A

Q12.In a frequency distribution, the class interval is 10 and the frequency of the median class is 15. If the cumulative frequency before the median class is 13, what is the median?

Correct Answer: B

Q13.In a continuous frequency distribution, the class interval with the highest frequency is known as the:

Correct Answer: C

Q14.If the mean of a dataset is 40 and the sum of all observations is 800, how many observations are there in the dataset?

Correct Answer: B

Q15.Francis Galton is known for pioneering the use of statistical methods in which field?

Correct Answer: A

Q16.In a frequency distribution, which class interval contains the median if the cumulative frequency just before it is 13 and the frequency of the median class is 15?

Correct Answer: B

Q17.What is the variance of a data set if the standard deviation is 5?

Correct Answer: D

Q18.If the mean of a data set is 20 and the sum of all observations is 200, how many observations are there?

Correct Answer: B

Q19.The range of a dataset is 50. If each observation is divided by 5, what will be the new range?

Correct Answer: A

Q20.In the context of statistics, what does the term 'dispersion' refer to?

Correct Answer: B

Q21.If the mean of a data set is 10 and the variance is 4, what is the standard deviation?

Correct Answer: A

Q22.In a continuous frequency distribution, the mid-point of a class interval is used to calculate which of the following?

Correct Answer: B

Q23.If the mean of a dataset is 100 and the standard deviation is 15, what will be the new standard deviation if each observation is multiplied by 3?

Correct Answer: C

Q24.If the variance of a dataset is 16, what will be the new variance if each data point is multiplied by 3?

Correct Answer: C

Q25.The variance of a data set is 16. If each observation is divided by 2, what will be the new variance?

Correct Answer: A

Q26.What is the mean deviation about the mean for the data set: 6, 7, 10, 12, 13, 4, 8, 12?