All questions of Elementary Statistics for RRB NTPC/ASM/CA/TA Exam

Please answer

To determine the class interval, we need to know the range of the data and the number of class intervals.

In this case, the class mark is given as 9.5 and the class size is 6. The class mark represents the midpoint of the class interval, which means that the lower limit of the class interval is 9.5 - 0.5 = 9 and the upper limit is 9.5 + 0.5 = 10.

Let's calculate the range of the data:
Range = Upper Limit - Lower Limit
Range = 10 - 9
Range = 1

Since the range is 1 and there are 6 class intervals, we can divide the range by the number of class intervals to determine the class interval size:
Class Interval Size = Range / Number of Class Intervals
Class Interval Size = 1 / 6
Class Interval Size = 0.1667

Now, we can determine the class intervals:
First Class Interval: 9 - 9.1667
Second Class Interval: 9.1667 - 9.3333
Third Class Interval: 9.3333 - 9.5
Fourth Class Interval: 9.5 - 9.6667
Fifth Class Interval: 9.6667 - 9.8333
Sixth Class Interval: 9.8333 - 10

As we can see, the class interval that includes the class mark of 9.5 is 9.5 - 9.6667. Therefore, the correct answer is option D) 6.5-12.5.

It is important to note that the other options given are incorrect because they do not include the class mark of 9.5 within the range of the class intervals.

Given data: 60, 58, 90, 51, 47, 81, 70, 95, 87, 99
Range = difference between the highest and the lowest value in the data
To find the range, we need to first arrange the data in ascending or descending order.

Arranging the data in ascending order:
47, 51, 58, 60, 70, 81, 87, 90, 95, 99

Range = highest value - lowest value
Range = 99 - 47
Range = 52

Therefore, the correct answer is option 'D' (52).

Option A is correct

**Solution:**

To find the difference between the modes when x = 9 and x = 8, we need to determine the modes for both scenarios and then calculate the difference between them.

Given data set: 7, 8, 8, 9, 9, x

**When x = 9:**

In this case, the data set becomes: 7, 8, 8, 9, 9, 9

The modes are the numbers that appear most frequently in the data set. In this case, the modes are 8 and 9, as they both appear twice.

**When x = 8:**

In this case, the data set becomes: 7, 8, 8, 9, 9, 8

The modes are the numbers that appear most frequently in the data set. In this case, the modes are 8 and 9, as they both appear twice.

Therefore, the modes are the same for both scenarios: 8 and 9.

**Calculating the difference between the modes:**

To calculate the difference between the modes, we subtract the smaller mode from the larger mode.

In this case, the smaller mode is 8 and the larger mode is 9.

Difference = 9 - 8 = 1

Therefore, the difference between the modes when x = 9 and x = 8 is 1.

Hence, the correct answer is option A.

Correct answer is option ‘ A ' because secondary data means data collected by someone else earlier like newspaper

To solve this problem, we need to use the concept of the mean and how it is affected by adding or removing numbers from a set of data. Let's break down the given information and solve the problem step by step.

Given Information:
- The mean of six numbers is 23.
- If one of the numbers is excluded, the mean of the remaining numbers becomes 20.

Step 1: Find the sum of the six numbers
Since the mean is the sum of all numbers divided by the total count, we can find the sum of the six numbers by multiplying the mean (23) by the count (6).
Sum of the six numbers = Mean * Count = 23 * 6 = 138

Step 2: Find the sum of the five remaining numbers
If the mean of the remaining five numbers is 20, we can find the sum of those numbers by multiplying the mean (20) by the count (5).
Sum of the five remaining numbers = Mean * Count = 20 * 5 = 100

Step 3: Find the excluded number
To find the excluded number, we need to subtract the sum of the five remaining numbers from the sum of the six numbers.
Excluded number = Sum of the six numbers - Sum of the five remaining numbers = 138 - 100 = 38

Conclusion:
The excluded number is 38 (option A) based on the given information and calculations above.

To find the lower limit of the class, we can subtract half of the width of the class from the mid-value of the class.

Given information:
Mid-value of the class = 60.5
Width of the class = 10

Finding the lower limit of the class:
Lower limit of the class = Mid-value of the class - (Width of the class / 2)

Substituting the given values:
Lower limit of the class = 60.5 - (10 / 2)

Simplifying the expression:
Lower limit of the class = 60.5 - 5

Therefore, the lower limit of the class is 55.5.

So, the correct answer is option 'A' (55.5).

Primary Data in Statistics

Primary data is the data that is collected firsthand by the researcher himself for a specific research purpose. It is original data that has not been processed or analyzed by anyone else. The researcher collects primary data by conducting surveys, experiments, observations, or interviews.

Explanation:

In the given scenario, a student collects information about the number of school going children in a locality consisting of a hundred households. This information is collected by the student himself for a specific research purpose, making it primary data. The data is collected through surveys or interviews conducted by the student.

Option C is the correct answer as it refers to primary data. Option A, arrayed data, refers to data that is arranged in a specific sequence or order. Option B, grouped data, refers to data that is organized into groups or categories for analysis. Option D, secondary data, refers to data that has already been collected and processed by someone else for a different research purpose.

Given, 0.25 cm represents 100 people.

To find the length of the bar that represents 2000 people, we need to use the concept of proportionality.

Let x be the length of the bar that represents 2000 people.

We know that,

0.25 cm represents 100 people

So,

x cm represents 2000 people

Using the concept of proportionality, we can write:

0.25/100 = x/2000

Simplifying the above equation, we get:

x = (0.25 x 2000)/100

x = 5 cm

Therefore, the length of the bar that represents 2000 people is 5 cm.

Hence, the correct option is (a) 5 cm.

First four prime number 2,3,5,7
mean = sum of all observation/Number of observation
= 2+3+5+7 / 4
= 17/4
= 4.25
so option D is correct

Explanation:

A grouped frequency distribution is a tabular representation of data that is divided into intervals or classes, and each class represents a range of values. In this case, the class intervals are 0-10, 10-20, 20-30, and so on.

The class width is the difference between the upper and lower class limits of a class interval. In this case, the lower class limit of the first interval is 0 and the upper class limit is 10. Therefore, the class width is:

Upper class limit - Lower class limit = 10 - 0 = 10

Hence, the correct answer is option A) 10.

The range of this data is the difference between the highest and lowest scores. In this case, the highest score is 99 and the lowest score is 47, so the range is 99-47=52. Therefore, the correct answer is (a) 52.

To solve this problem, we need to understand the concept of class limits and how they relate to the width of each class.

Class limits are the minimum and maximum values within a class interval. The lower class limit is the smallest value within a class, while the upper class limit is the largest value within a class.

Given that the width of each class is 5, it means that the difference between the upper class limit and the lower class limit is 5. Therefore, if we know the lower class limit of the lowest class, we can determine the upper class limit of the highest class.

Let's consider the given information:

- Width of each class: 5
- Lower class limit of the lowest class: 10

To find the upper class limit of the highest class, we need to determine the difference between the lower class limit of the highest class and the width of each class.

Let's denote the lower class limit of the highest class as "L" and find its value:

L = Lower class limit of the lowest class + (Number of classes - 1) × Width of each class

Number of classes can be calculated using the formula:

Number of classes = (Range of data) / (Width of each class)

The range of data is the difference between the maximum and minimum values in the dataset. However, since we don't have this information, we cannot directly calculate the number of classes.

However, we do know that the lower class limit of the lowest class is 10. Therefore, we can make an assumption that the minimum value in the dataset is 10. This assumption allows us to calculate the number of classes:

Number of classes = (Range of data) / (Width of each class)
= (Maximum value - Minimum value) / (Width of each class)
= (L - 10) / 5

Since we know that the width of each class is 5, we can rearrange the equation to solve for L:

L = 10 + (Number of classes - 1) × Width of each class
= 10 + (((L - 10) / 5) - 1) × 5

Simplifying the equation:

L = 10 + (L - 10) - 5
L = L - 10 - 5 + 10
L = L - 5

At this point, we can see that L is equal to the upper class limit of the highest class. Therefore, the answer is option B) 35.

To find the median, we need to determine the middle value of the observations. In this case, we have 16 observations arranged in ascending order.

Given that the 8th observation is 25 and the 9th observation is 27, we can see that these two values are consecutive. This means that there are 7 observations before the 8th observation (including the 8th observation itself) and 6 observations after the 9th observation (including the 9th observation itself).

Since there are an even number of observations (16), the median will be the average of the two middle values. In this case, the two middle values are the 8th and 9th observations, which are 25 and 27.

To find the average, we add the two values together and divide by 2:
(25 + 27) / 2 = 52 / 2 = 26.

Therefore, the median of the given observations is 26.

Hence, the correct answer is option A) 26.

Class width is the same as the class size. The class size of the given intervals 1-20, 21-40, 41-60,.. is 20.

Mode:
The mode of a set of numbers is the value or values that appear most frequently. In other words, it is the number(s) that occur(s) the highest number of times in the set.

Given Set:
4, 6, 7, 8, 12, 11, 13, 9, 13, 9, 7, 8, 9

To find the mode of this set, we need to determine which number(s) occur(s) most frequently.

Frequency Count:
To count the frequency of each number in the set, we can create a frequency table which lists each number along with its corresponding frequency.

Number | Frequency
-------|----------
4 | 1
6 | 1
7 | 2
8 | 2
9 | 3
11 | 1
12 | 1
13 | 2

From the frequency table, we can see that the number 9 appears the most frequently, with a frequency of 3. Therefore, the mode of the given set is 9.

Answer:
The mode of the given set {4, 6, 7, 8, 12, 11, 13, 9, 13, 9, 7, 8, 9} is 9.

26

To determine which variables are discrete, we need to understand the characteristics of discrete variables.

A discrete variable is a variable that can only take on specific values, often whole numbers, and has gaps or intervals between the values.

Now let's analyze each variable given in the options:

1. Size of shoes:
- The size of shoes is usually measured in whole numbers, such as 6, 7, 8, etc.
- It cannot take on decimal values or be measured continuously.
- Therefore, the size of shoes is a discrete variable.

2. Number of pages in a book:
- The number of pages in a book is also measured in whole numbers, such as 100, 200, 300, etc.
- It cannot have decimal values or be measured continuously.
- Therefore, the number of pages in a book is a discrete variable.

3. Distance travelled by a train:
- The distance travelled by a train can have any value along a continuous scale, such as 10.5 km, 25.3 km, etc.
- It can take on any value within a given range, without any gaps or intervals.
- Therefore, the distance travelled by a train is a continuous variable, not discrete.

4. Time:
- Time can be measured in hours, minutes, seconds, etc.
- It can take on any value within a given range, without any gaps or intervals.
- Therefore, time is a continuous variable, not discrete.

Based on the characteristics of discrete variables, the variables that are discrete in the given options are:
- Size of shoes
- Number of pages in a book

Therefore, the correct answer is option A: 1 and 2.

**Range of Data**

The range of a given set of data refers to the difference between the maximum and minimum observations in that data set. It is a measure of the spread or dispersion of the data points. In other words, it measures the extent to which the data values are scattered or varied.

**Calculation of Range**

To calculate the range of a data set, you need to find the difference between the maximum and minimum values. This can be done by following these steps:

1. Identify the maximum value in the data set.
2. Identify the minimum value in the data set.
3. Subtract the minimum value from the maximum value to find the range.

For example, consider the following set of data: 5, 10, 12, 8, 15, 3

The maximum value is 15 and the minimum value is 3. Therefore, the range can be calculated as:

Range = Maximum value - Minimum value
= 15 - 3
= 12

Hence, the range of the given data set is 12.

**Importance of Range**

The range provides a simple and straightforward measure of the spread of data. It gives an idea of how much the data values deviate from each other. A larger range indicates a greater dispersion, while a smaller range indicates a smaller dispersion.

By calculating the range, we can quickly determine the minimum and maximum values in a data set. This information is useful in various statistical analyses and interpretations. For example, it can help identify outliers (data points that are significantly different from the rest of the data) and determine the overall variability of the data.

Moreover, the range is a basic measure of dispersion and is often used as a starting point for more advanced statistical analyses, such as calculating the standard deviation or variance.

In conclusion, the difference between the maximum and minimum observations in a data set is known as the range. It provides a measure of the spread or dispersion of the data and is a fundamental concept in statistics.

And so on, what is the midpoint of the class interval 31-40?

The midpoint of a class interval is found by adding the lower and upper limits of the interval and dividing by 2.

For the interval 31-40, the lower limit is 31 and the upper limit is 40.

Midpoint = (31 + 40) / 2 = 71 / 2 = 35.5

Therefore, the midpoint of the class interval 31-40 is 35.5.

**Explanation:**

**Secondary data** is the data that is collected by someone else for their own purpose. In this case, the data is collected from a newspaper. Let's understand why the given data is considered secondary data.

**1. Definition of Secondary Data:**
Secondary data refers to the data that is collected by someone other than the user for their own purpose. It is not collected specifically for the purpose of the current analysis.

**2. Collection of Data from Newspapers:**
In this scenario, the data is collected from a newspaper. Newspapers are a common source of information for various topics, including election results. The newspaper collects the data for its own purpose, which is to inform the public about the election results.

**3. Characteristics of Secondary Data:**
The data collected from newspapers possesses the following characteristics of secondary data:

a) **Collected by someone else:** The data is collected by the newspaper, not by the analyst or researcher conducting the election analysis.

b) **Not collected for the current analysis:** The data is collected by the newspaper for the purpose of reporting the election results to the public, not specifically for the purpose of conducting an analysis.

c) **Readily available:** Newspapers publish election results for public consumption, making the data readily available for analysis.

**4. Usage of Secondary Data:**
Secondary data is used by researchers and analysts for various purposes, including:

a) **Comparison:** Researchers can compare the findings of their analysis with the data collected from other sources, such as newspapers, to validate their findings.

b) **Historical analysis:** Secondary data collected by newspapers over time can be used for historical analysis, studying trends and patterns in election results.

c) **Supplementary information:** Secondary data can provide additional information that complements primary data collected through surveys or interviews.

In conclusion, the data collected from newspapers to analyze election results is considered secondary data because it is collected by someone else (newspaper) for their own purpose (informing the public) and not specifically for the current analysis.

We have learnt that Lowest observation - highest observation = range. thats why range is correct answer

Discrete variables are those that can only take on specific, distinct values. They cannot take on any value between two specific points.

In the given options, the following variables are discrete:

1. Size of shoes - This is a discrete variable as shoe sizes come in specific, whole numbers such as 6, 7, 8, and so on. There are no values between 6 and 7 or 7 and 8, etc.

2. Number of pages in a book - This is also a discrete variable as the number of pages in a book can only be a whole number, such as 100, 200, 300, etc.

The following variables are not discrete:

3. Distance travelled by a train - This is a continuous variable as the distance travelled by a train can take on any value between two specific points, such as 10.5 miles, 15.3 miles, etc.

4. Time - This is also a continuous variable as time can take on any value between two specific points, such as 2:30 pm, 2:31 pm, etc.

Therefore, the correct answer is option D, which includes variables 1 and 2, both of which are discrete variables.

Given:
- Mid-value of the class = 60.5
- Width of the class = 10

To Find:
- Lower limit of the class

Solution:
To find the lower limit of the class, we need to subtract half of the width from the mid-value of the class.

Step 1: Calculate half of the width of the class.
Half of the width = 10/2 = 5

Step 2: Subtract half of the width from the mid-value of the class.
Lower limit = Mid-value - Half of the width
Lower limit = 60.5 - 5
Lower limit = 55.5

Therefore, the lower limit of the class is 55.5.

Answer: Option A (55.5)

The class corresponding to the class mark 43 is b) 38.5-48.5.

To determine the class corresponding to a given class mark, we need to look at the intervals between the class marks. In this case, the intervals are:

- 38-43
- 43-48
- 48-53
- 53-58

We can see that the class mark 43 falls within the interval 38-43 and the corresponding class is b) 38.5-48.5. This is because the interval is halfway between 38 and 48, so we add 0.5 to each end to get 38.5-48.5.

Explanation:

Given data:
Minimum value = 82
Range = 38

Range is the difference between the maximum and minimum values of a data set. Therefore, if we add the range to the minimum value, we can find the maximum value of the data set.

Using the formula, Maximum Value = Minimum Value + Range, we get:

Maximum Value = 82 + 38
Maximum Value = 120

Therefore, the maximum value of the data set is 120, which is option D.

Answer: Option D (120)

Mean of Observations

The mean of a set of observations is the sum of the observations divided by the total number of observations.

Formula for Mean: (Sum of Observations) / (Total number of Observations)

Given Observation Set

The observation set is: 4, 7, x, 8, 9, 10.

Mean of Observation Set

The mean of the observation set is given as 8.

Therefore, (4+7+x+8+9+10) / 6 = 8.

4+7+x+8+9+10 = 48.

x = 48-38 = 10.

Hence, the value of x is 10.

Answer

Therefore, the correct answer is option 'D' which is 10.

Finding the Class Mark of a Class Interval

To find the class mark of a class interval, we need to determine the midpoint of the interval. The class mark is also known as the class midpoint or the class center. It is the average of the lower and upper limits of the class interval.

Formula for Class Mark

Class Mark = (Lower Limit + Upper Limit) / 2

Calculation

The given class interval is 2.4-6.6.

Lower Limit = 2.4
Upper Limit = 6.6

Class Mark = (Lower Limit + Upper Limit) / 2
Class Mark = (2.4 + 6.6) / 2
Class Mark = 9 / 2
Class Mark = 4.5

Therefore, the class mark of the class interval 2.4-6.6 is 4.5.

Answer: Option C

Explanation:

The mean of a set of numbers is the sum of all the numbers in the set divided by the total number of numbers in the set.

In this case, we are looking for the mean of counting numbers through 100, which means we need to find the sum of all the numbers from 1 to 100 and divide that sum by the total number of numbers in the set (which is 100).

Calculating the sum of counting numbers through 100:
We can use the formula for the sum of an arithmetic series to find the sum of counting numbers through 100:

S = (n/2)(a + l)

where S is the sum, n is the number of terms, a is the first term, and l is the last term.

In this case, n = 100, a = 1, and l = 100, so we can plug in these values and simplify:

S = (100/2)(1 + 100)
S = 50(101)
S = 5050

So the sum of counting numbers through 100 is 5050.

Calculating the mean:
Now that we have the sum, we can find the mean:

mean = sum/number of terms
mean = 5050/100
mean = 50.5

Therefore, the mean of counting numbers through 100 is 50.5.

Explanation:
To determine the class marks of the first two class intervals, we need to understand what class marks are and how they are calculated.

Class Size:
The class size refers to the number of observations or data points in a given distribution. In this case, the class size is 25.

Class Interval:
A class interval represents a range of values in a data set. Each class interval is defined by its lower limit and upper limit. In this case, the first class interval is 200-224.

Class Mark:
The class mark is the average of the upper and lower limits of a class interval. It represents the central value of the interval. The class mark is calculated using the formula:
Class Mark = (Lower Limit + Upper Limit) / 2

Determining the Class Marks:
To find the class marks of the first two class intervals, we need to calculate the average of the upper and lower limits of each interval.

The first class interval is 200-224.
Lower Limit = 200
Upper Limit = 224
Class Mark = (200 + 224) / 2 = 424 / 2 = 212

Therefore, the class mark of the first class interval is 212.

To find the class mark of the second class interval, we need to determine the upper and lower limits of the interval.

Since the class size is 25 and the first class interval has a range of 25 (224 - 200 = 25), we can determine the upper limit of the second class interval as follows:
Upper Limit of Second Class Interval = Upper Limit of First Class Interval + Range of Class Interval
Upper Limit of Second Class Interval = 224 + 25 = 249

Therefore, the second class interval is 224-249.

Lower Limit = 224 (Upper Limit of First Class Interval)
Upper Limit = 249
Class Mark = (224 + 249) / 2 = 473 / 2 = 236.5 ≈ 237

Therefore, the class mark of the second class interval is 237.

Conclusion:
The class marks of the first two class intervals are 212 and 237, which corresponds to option 'D'.

D.Grouped data

The correct answer is option 'B' - Range.

Central tendency is a statistical measure used to describe the center or typical value of a data set. It provides a single value that represents the entire data set. Common measures of central tendency include the mode, median, and mean.

- Mode: The mode is the value that appears most frequently in a data set. It is the only measure of central tendency that can be used for categorical data. For example, if we have the data set {1, 2, 2, 3, 4, 4, 4, 5}, the mode is 4 because it appears most frequently.

- Median: The median is the middle value of a data set when it is arranged in ascending or descending order. It is not affected by extreme values or outliers. For example, in the data set {1, 2, 3, 4, 5}, the median is 3.

- Mean: The mean is the average of a data set. It is calculated by summing all the values and dividing the sum by the number of values. For example, in the data set {1, 2, 3, 4, 5}, the mean is (1+2+3+4+5)/5 = 3.

- Range: Range is not a measure of central tendency. It is a measure of dispersion that represents the difference between the highest and lowest values in a data set. For example, in the data set {1, 2, 3, 4, 5}, the range is 5-1 = 4.

The range provides information about the spread or variability of the data, but it does not indicate the central or typical value. It is useful for understanding the overall range of the data but does not provide a single value that represents the center.

In conclusion, the range is not a common measure of central tendency. The mode, median, and mean are commonly used to describe the central or typical value of a data set.

Understanding the Mode
The mode is a fundamental concept in statistics that represents the most frequently occurring value in a set of observations. Here's a detailed explanation of why option 'B' is correct:
Definition of Mode
- The mode is defined as the value that appears most often in a data set.
- Unlike other measures of central tendency like mean and median, the mode does not require the data to be numerical or continuous.
Characteristics of the Mode
- Multimodal Data: A set of observations can have more than one mode (bimodal or multimodal) if multiple values occur with the same highest frequency.
- Non-Numeric Data: The mode can be applied to categorical data, making it versatile in various fields such as marketing and social sciences.
Comparison with Other Measures
- Sum of Observations: This is not relevant to mode; it refers to the total of all values.
- Mean of Middle Two Observations: This describes the median, not the mode.
- Divides Observations: This is a characteristic of the median, which divides a data set into two equal halves.
Importance of Mode
- Data Insight: The mode provides insights into the most common values in a data set, which can be useful for understanding trends and patterns.
- Simplicity: It is easy to determine and interpret, making it a useful statistic for quick analysis.
In summary, the mode effectively represents the most common observation within a data set, confirming that option 'B' is indeed the correct answer.