Odisha State Board CHSE Odisha Class 12 Education Solutions Chapter 19 Frequency Distribution Questions and Answers.
CHSE Odisha 12th Class Education Chapter 19 Question Answer Frequency Distribution
Group – A
Short type Questions with Answers
I. Answer with in Two/Three sentence :
Question 1.
What is a frequency distribution, and why is it used in statistics?
Answer:
A frequency distribution is a tabular representation of statistical data that lists the values of a variable along with their respective frequencies. It is used to organize and summarize data, providing insights into the distribution and patterns within the dataset.
Question 2.
What is the difference between ungrouped and grouped frequency distributions?
Answer:
An ungrouped frequency distribution lists individual values and their frequencies, suitable for a small set of data. In contrast, a grouped frequency distribution organizes data into intervals or classes, making it more practical for larger datasets.
Question 3.
How can class intervals be determined in a frequency distribution?
Answer:
Class intervals should be clearly defined, not too large or too small, and of the same width. The width of each interval is calculated using the formula: Width = Range / Number of classes.
Question 4.
Why should open-ended cases in class intervals be avoided in a frequency distribution?
Answer:
Open-ended cases, where there is polower limit of the first group or no upper limit of the last group, should be avoided because they create difficulties in analysis and interpretation, hindering the accurate representation of data.
Question 5.
What is the significance of having a suitable number of classes in a frequency distribution?
Answer:
The number of classes should be neither too large nor too small; typically, between 6 and 15 classes are considered adequate. Fewer classes result in wider intervals, leading to loss of accuracy, while too many classes complicate the distribution.
Question 6.
How is cumulative frequency related to a frequency distribution?
Answer:
Cumulative frequency represents the sum of consecutive frequencies in a frequency distribution. It provides information about the total number of units below or above specified values of class intervals.
Question 7.
Why should class intervals be continuous throughout a frequency distribution?
Answer:
Continuous intervals ensure that every data point falls into precisely one class, avoiding misrepresentation. For example, age groups like 20-24 and 25-29 are more representative than 24-25 and 29-30.
Question 8.
What is the purpose of a less than cumulative frequency distribution?
Answer:
A less than cumulative frequency distribution focuses on determining the number of units below a specified upper limit of a class interval, aiding in analyzing data distribution.
Question 9.
How is the width of class intervals calculated in a frequency distribution?
Answer:
The width of class intervals is determined by dividing^the range of data (difference between highest and lowest values) by the number of classes.
II. Answer with in Five/Six sentence :
Question 1.
What is a frequency distribution, and how does it organize statistical data?
Answer:
A frequency distribution organizes statistical data by listing the values of a variable and their corresponding frequencies in a tabular form. It provides a clear summary of how often each value occurs in the data set. For instance, if we survey 20 families to determine the number of children they have, the resulting raw data can be organized into an ungrouped frequency distribution by listing the frequencies against each unique value.
Question 2.
What are the guidelines for constructing a frequency distribution?
Answer:
Constructing a frequency distribution involves defining clear classes, determining an appropriate number of classes (usually between 6 and 15), ensuring uniform class width, avoiding open-ended cases, and maintaining continuity throughout the distribution. The lower limits of class intervals should be simple multiples of the interval width for simplicity in construction and interpretation. These guidelines contribute to a meaningful and accurate representation of the data.
Question 3.
How can class intervals be determined in a frequency distribution?
Answer:
Class intervals in a frequency distribution are determined by considering the range of data (difference between the highest and lowest values) and the number of desired classes. The formula for the width of each interval is Width = Range / Number of classes. This ensures that the intervals are of equal width, facilitating easy computation and interpretation.
Question 4.
What is the importance of cumulative frequency in a distribution?
Answer:
Cumulative frequency provides information about the total number of units below or above specified values in a distribution. Less than cumulative frequency distribution focuses on the number of items below a specified upper limit, while greater than cumulative frequency distribution focuses on the number of cases above a specified lower limit. It aids in understanding the distribution’s cumulative impact and is useful for various analytical purposes.
Question 5.
Why are relative frequency and percentage frequency useful in statistical analysis?
Answer:
Relative frequency, obtained by dividing the frequency in each class by the total number of observations, allows for comparing distributions of different sizes. Percentage frequency, derived from relative frequency by multiplying by 100, expresses the proportion of cases in each group. These measures are particularly helpful in meaningful comparisons when dealing with unequal sample sizes, as percentages make the comparison independent of the total number of case.
Question 6.
What is the purpose of cumulative relative frequency distribution?
Answer:
Cumulative relative frequency distribution indicates the proportion or percentage of cases falling below or above a specific score point. A less than cumulative relative frequency distribution shows the proportion of cases below the upper limit of a class interval, while a greater than cumulative relative frequency distribution shows the proportion above the lower limit. These distributions provide valuable insights into the cumulative distribution pattern of the data.
Question 7.
How is a percentage frequency distribution calculated, and why is it beneficial?
Answer:
A percentage frequency distribution is obtained by multiplying each relative frequencyby 100. It expresses the proportion of cases in each class interval as a percentage of the total. This is beneficial for comparing distributions, as percentages make the comparison independent of the total number of cases. In the context of the example, it helps understand the distribution of the number of children in families as a percentage of the total surveyed.
Question 8.
Why should open-ended cases in constructing frequency distributions be avoided?
Answer:
Open-ended cases, where there is no lower limit for the first group or no upper limit for the last group, should be avoided in constructing frequency distributions. This avoidance is crucial because open-ended cases create difficulties in analysis and interpretation. Clearly defined class intervals with both lower and upper limits ensure accuracy and consistency in representing the data.
Question 9.
What role does class interval width play in constructing a frequency distribution?
Answer:
Class interval width, determined by the range of data and the number of classes, plays a crucial role in constructing a frequency distribution. It influences the clarity of representation, ease of computation, and the overall accuracy of the distribution. Uniform class interval width is preferred for simplifying calculations and facilitating a meaningful interpretation of the data.
Question 10.
In what scenarios can a relative frequency distribution be particularly helpful?
Answer:
Relative frequency distributions are particularly helpful when comparing two or more distributions with different sample sizes. By expressing the frequency of each class as a proportion of the total, relative frequency allows for a fair comparison regardless of the overall size of the samples or populations under consideration. This makes it a valuable tool for researchers and analysts seeking meaningful comparisons in diverse data sets.
Group – B
Long Type Questions With Answers
Question 1.
How is a frequency distribution constructed?
Answer:
Statistical data can be organized into a frequency distribution which simply lists the value of the variable and frequency of its occurrence in a tabular form. A frequency distribution can be defined as the list of all the values obtained in the data and the corresponding frequency with which these values occur in the data. The frequency distribution can either be ungrouped or grouped. When the number of values of the variable is small, then we can construct an ungrouped frequency distribution which is simply listing the frequency of occurrence against the value of the given variable. As an example, let ns assume that 20 families were surveyed to find out how many children each family had. The raw data obtained from the survey is as follows: 0,2,3,1,1,3,4,2,0,3,4,2,2,1,0,4,1,2,2,3
This data can be classified into an ungrouped frequency distribution. The number of children becomes our variable (X) for which we can list the frequency of occurrence (f) in a tabular form as follows:
Constructing a Frequency Distribution :
| Age Group (Years) | Frequency |
| 20 to less than 25 | 5 |
| 25 to less than 30 | 15 |
| 30 to less than 35 | 25 |
| 35 to less than 40 | 30 |
| 40 to less than 45 | 15 |
| 45 to less than 50 | 10 |
| Total | 100 |
The number of groups and the size of class interval are more or less arbitrary in nature within the general guidelines established for constructing a frequency distribution. The following guidelines for such a construction may be considered:
(i) The classes should be clearly defined, and each of the observations should be included in only one of the class intervals. This means that the intervals should be chosen in such a manner that one score cannot belong to more than one class interval, so that there is no overlapping of class intervals.
(ii) The number of classes should neither be too large nor too small. Normally, between 6 and 15 classes are considered to be adequate. Fewer class intervals would mean a greater class interval width with consequent loss of accuracy. Too many class intervals result in a greater complexity.
(iii) All intervals should be of the same width. This is preferred for easy computations. A suitable class width can be obtained by knowing the range of data (which is the absolute difference between the highest value and the lowest value in the data) and the number of classes which are predetermined, so that: The width of the interval = Range / Number of classes In the case of ages of factory workers where the youngest worker was 20 years old and the oldest was 50 years old, the range would be 50-20 = 30. If we decide to make 10 groups then the width of each class would be: 30/10 = 3 Similarly, if we decide to make 6 classes instead of 10, then the width of each class interval would be: 30/6 = 5
(iv) Open-ended cases where there is no lower limit of the first group or no upper limit of the last group should be avoided since this creates difficulty in analysis and interpretation. (The lower and upper values of a class interval are known as lower and upper limits.)
(v) Intervals should be continuous throughout the distribution. For example, in the case of factory workers, we could group them in groups of 20 to 24 years, then 25 to 29 years, and so on, but it would be highly misleading because it does not accurately represent the person who is between 24 and 25 years or between 29 and 30 years, and so on. Accordingly, it is more representative to group them as: 20 years to less than 25 years, 25 years to less than 30 years. In this way, everybody who is 20 years and a fraction less than 25 years is included in the first category and the person who is exactly 25 years and above but a fraction less than 30 years would be included in the second category, and so on. This is especially important for continuous distributions.
(vi) The lower limits of class intervals should be simple multiples of the interval width. This is primarily for the purpose of simplicity in construction and interpretation. In our example of 20 years but less than 25 years, 25 years but less than 30 years, and 30 years but less than 35 years, the lower limit values for each class are simple multiples of the class width which is 5
Group – C
Objective type Questions with Answers
I. Multiple Choice Quotums with Answers
Question 1.
What is a frequency distribution?
(i) A summary of raw data
(ii) A collection of unorganized data
(iii) A representation Of qualitative information
(iv) An individual data point
Answer:
(i) A summary of raw data
Question 2.
In constructing a frequency distribution, when is an ungrouped frequency distribution suitable?
(i) When the number of values’is large
(ii) When the data is continuous
(iii) When the number of values is small
(iv) When the data is categorical
Answer:
(iii) When the number bf values is small
Question 3.
What guidelines are considered for determining the number of classes in a frequency distribution?
(i) Between 1 and 5 classes are considered adequate
(ii) Between 6 and 15 classes are considered adequate
(iii) As many classes as possible for accuracy
(iv) No specific guidelines for the number of classes
Answer:
(ii) Between 6 and 15 classes are considered adequate
Question 4.
How is the width of a class interval determined in a frequency distribution?
(i) It is arbitrary and depends on personal preference
(ii) Width = Range / Number of classes
(iii) It is always the same for all distributions
(iv) It is irrelevant in frequency distribution construction
Answer:
(ii) Width = Range / Number of classes
Question 5.
Why should open-ended cases in class intervals be avoided in frequency distribution?
(i) They provide more flexibility in analysis
(ii) They simplify computations
(iii) They create difficulties in analysis and interpretation
(iv) They are necessary for continuous distributions
Answer:
(iii) They create difficulties in analysis and interpretation
Question 6.
What is cumulative frequency in a frequency distribution?
(i) The total number of units in each class interval
(ii) The sum of consecutive frequencies
(iii) The highest frequency in the distribution
(iv) The average frequency per class
Answer:
(ii) The sum of consecutive frequencies
Question 7.
What does a less than cumulative frequency distribution represent?
(i) The number of cases above a specified value
(ii) The proportion of cases below the upper limit of a class interval
(iii) The percentage of cases falling between two values
(iv) The average frequency in a distribution
Answer:
(ii) The proportion of cases below the upper limit of a class interval
Question 8.
What does a greater than cumulative frequency distribution represent?
(i) The total number of cases in the distribution
(ii) The proportion of cases above the lower limit of a class interval
(iii) The cumulative frequency of the highest class interval
(iv) The percentage of cases falling between two values
Answer:
(ii) The proportion of cases above the lower limit of a class interval
Question 9.
What does a relative frequency distribution represent?
(i) The total number of observations in a distribution
(ii) The proportion of cases in each class relative to the total
(iii) The sum of frequencies in a distribution
(iv) The cumulative frequency of the highest class interval
Answer:
(ii) The proportion of cases in each class relative to the total
Question 10.
How is a percentage frequency calculated in a frequency distribution?
(i) By dividing the frequency by the total number of observations
(ii) By multiplying the frequency by the class width
(iii) By dividing the cumulative frequency by the total number of observations
(iv) By multiplying the relative frequency by 100
Answer:
(iv) By multiplying the relative frequency by 100
II. Fill in the blanks :
Question 1.
Statistical data can be organized into a _____ distribution, which lists the value of the variable and the frequency of its occurrence in a tabular form.
Answer:
Frequency
Question 2.
When the number of values is small, an _____ frequency distribution can be constructed by listing the frequency of occurrence against the value of the given variable.
Answer:
Ungrouped
Question 3.
The width of each class interval in a frequency distribution can be calculated using the formula: Width of the interval = _____
Answer:
Range / Number of classes
Question 4.
In constructing a frequency distribution, it is important that all intervals should be of _____.
Answer:
The same width
Question 5.
The lower and upper values of a class interval are known as _____ and _____ respectively.
Answer:
Lower limits, Upper limits
Question 6.
Intervals in a frequency distribution should be continuous throughout to accurately represent individuals within each _____.
Answer:
Category
Question 7.
Cumulative frequency distribution helps determine the total number of units that lie or _____ the _____ specified values of class intervals.
Answer:
Below, Above
Question 8.
The cumulative frequency distribution with interest in the number of items below a specified value is known as a _____ cumulative frequency distribution.
Answer:
Less than
Question 9.
Percentage frequency distribution is obtained by multiplying each relative frequency by _____
Answer:
100
Question 10.
A _____ relative frequency distribution shows the proportion of cases lying below the upper limit of a specific class interval:
Answer:
Less than
III. Answer the following questions in one word:
Question 1.
What is a frequency distribution?
Answer:
A frequency distribution is a tabular representation of statistical data, listing values of a variable along with the corresponding frequencies of their occurrence.
Question 2.
How is an ungrouped frequency distribution different from a grouped frequency distribution?
Answer:
An ungrouped frequency distribution lists frequencies for individual values, while a grouped frequency distribution categorizes values into intervals and lists the frequencies for each interval.
Question 3.
What is the purpose of constructing a frequency distribution?
Answer:
The purpose of constructing a frequency distribution is to organize and summarize data, making it easier to analyze and interpret patterns.
Question 4.
What is the range of data, and how is it calculated?
Answer:
The range of data is the absolute difference between the highest and lowest values. It is calculated as Range = Highest Value – Lowest Value.
Question 5.
Why is it important to avoid open-ended cases in constructing a frequency distribution?
Answer:
Open-ended cases create difficulty in analysis and interpretation as they lack either a lower limit for the first group or an upper limit for the last group.
Question 6.
What does the term “cumulative frequency” represent in a distribution?
Answer:
Cumulative frequency represents the running total of frequencies in a distribution up to a specific point, either less than or greater than a specified value.
Question 7.
How is the width of a class interval determined in constructing a frequency distribution ? Answer: The width of a class interval is determined by dividing the range of data by the number of classes: Width = Range / Number of Classes.
Question 8.
Why should intervals be continuous in a frequency distribution?
Answer:
Continuous intervals ensure that each observation is accurately represented, avoiding misleading representations in the distribution.
Question 9.
What is the purpose of a cumulative relative frequency distribution?
Answer:
A cumulative relative frequency distribution provides the proportion or percentage of cases below or above specific score points in a distribution.
Question 10.
How is the percentage frequency calculated in a relative frequency distribution?
Answer:
Percentage frequency is calculated by multiplying each relative frequency by 100 in a relative frequency distribution.
Statistical data can be organized into a frequency distribution which simply lists the value of the variable and frequency of its occurrence in a tabular form. A frequency distribution , can be defined as the list of all the values obtained in the data and the corresponding frequency with which these values occur in the data.
The frequency distribution can either be ungrouped or grouped. When the number of values of the variable is small, then we can construct an ungrouped frequency distribution which is simply listing the frequency of occurrence against the value of the given variable. As an example, let us assume that 20 families were surveyed to find out how many children each family had. The raw data obtained from the survey is as follows: 0,2,3, 1,1,3,4,2,0,3,4,2,2,1,0,4,1,2,2,3
This data can be classified into an ungrouped frequency distribution. The number of children becomes our variable (X) for which we can list the frequency of occurrence (f) in a tabular form as follows:
Constructing a Frequency Distribution :
| Age Group (Years) | Frequency |
| 20 to less than 25 | 5 |
| 25 to less than 30 | 15 |
| 30 to less than 35 | 25 |
| 35 to less than 40 | 30 |
| 40 to less than 45 | 15 |
| 45 to less than 50 | 10 |
| Total | 100 |
The number of groups and the size of class interval are more or less arbitrary in nature within the general guidelines established for constructing a frequency distribution. The following guidelines for such a construction may be considered:
(i) The classes should be clearly defined and each of the observations should be included in only one of the class intervals. This means that the intervals should be chosen in such a manner that one score cannot belong to more than one class interval, so that there is no overlapping of class intervals.
(ii) The number of classes should neither be too large nor too small. Normally, between 6 and 15 classes are considered to be adequate. Fewer class intervals would mean a greater class interval width with consequent loss of accuracy. Too many class intervals result in a greater complexity.
(iii) All intervals should be of the same width. This is preferred for easy computations.’A suitable class width can be obtained by knowing the range of data (which is the absolute difference between the highest value and the lowest value in the data) and the number of classes which are predetermined, so that:
Range Number of classes In the case of ages of factory workers where the youngest worker was 20 years old and the oldest was 50 years old, the range would be 50-20 = 30. If we decide to make 10 groups then the width of each class would be: 30/10 = 3 Similarly, if we decide to make 6 classes instead of 10, then the width of each class interval would be: 30/6 = 5.
(iv) Open-ended cases where there is no lower limit of the first group or no upper limit of the last group should be avoided since this creates difficulty in analysis and interpretation. (The lower and upper values of a class interval are known as lower and upper limits.)
(v) Intervals should be continuous throughout the distribution. For example, in the case of factory workers, we could group them in groups of 20 to 24 years, then 25 to 29 years, and so on, but it would be highly misleading because it does not accurately represent the person who is between 24 and 25 years or between‘29 and 30 years, and so on. Accordingly, it is more representative to group them as: 20 years to less than 25 years, 25 years to less than 30 years. In this way, everybody who is 20 years and a fraction less than 25 years is included in the first category and the person who is exactly 25 years and above but a fraction less than 30 years would be included in the second category, and so on. This is especially important for continuous distributions.
(vi) The lower limits of class intervals should be simple multiples of the interval width. This is primarily for the purpose of simplicity in construction and interpretation. In our example of 20 years but less than 25 years, 25 years but less than 30 years, and 30 years but less than 35 years,, the lower limit values for each class are simple multiples of the class width which is 5.
Cumulative Frequency:
While the frequency distribution table tells us the number of units in each class interval, it does not tell us directly the total number of units that lie below or above the specified values of class intervals. This can be determined from a cumulative frequency distribution. When the interest of the investigator focusses on the number of items below a specified value, then this specified value is the upper limit of the class interval. It is known as less than cumulative frequency distribution. Similarly, when the interest lies in finding the number of cases above a specified value, then this value is taken as the lower limit of the specified class interval and is known as more than cumulative frequency distribution. The cumulative frequency simply means summing up the consecutive
| Class Interval (Years) | Frequency (f) | Cumulative Frequency (Greater Than) |
| 15 and up to 25
25 and up to 35 35 and up to 45 45 and up to 55 55 and up to 65 65 and up to 75 |
5
3 7 5 3 7 |
30 (greater than 15)
25 (greater than 25) 22 (greater than 35) 15 (greater than 45) 10 (greater than 55) 7 (greater than 65) |
| Class Interval (Years) | Frequency (f) | Cumulative Frequency (Greater Than) |
| 15 and up to 25
25 and up to 35 35 and up to 45 45 and up to 55 55 and up to 65 65 and up to 75 |
5
3 7 5 3 7 |
30 (greater than 15)
25 (greater than 25) 22 (greater than 35) 15 (greater than 45) 10 (greater than 55) 7 (greater than 65) |
Percentage Frequency :
The frequency distribution, as defined earlier, is a summary table in which the original data is condensed into groups and their frequencies. But if a researcher would like to know the proportion or the percentage of cases in each group, instead of simply the number of cases, he can do so by constructing a relative frequency distribution table. The relative frequency distribution can be formed by dividing the frequency in each class of the frequency distribution by the total number of observations.
It can be converted into a percentage frequency distribution by simply multiplying each relative frequency by 100. The relative frequencies are particularly helpful when comparing two or more frequency distributions in which the number of cases under investigation is not equal. The percentage distributions make such a comparison more meaningful, since percentages are relative frequencies and hence the total number in the sample or population under consideration becomes irrelevant. Carrying on with the earlier example:
| Class Interval (Years) | Frequency (f) | Relative Frequency (Rel. Freq.) | Percentage Frequency (% Freq.) |
| 15 and up to 25
25 and up to 3535 and up to 45 45 and up to 55 55 and up to 65 65 and up to 75 |
5
3 7 5 3 7 |
5/30 = 1/6
3/30 = 1/10 7/30 = 7/30 5/30 = 1/6 3/30 = 1/10 7/30 = 7/30 |
16.7
10.0 23.3 16.7 10.0 23.3 |
| Total | 30 | 1.0 | 100.0 |
Cumulative relative frequency distribution :
It is often useful to know the proportion or the percentage of cases falling below a particular score point or falling above a particular score point. A less than cumulative relative frequency distribution shows the proportion of cases lying below the upper limit of specific class interval. Similarly, a greater than cumulative frequency distribution shows the proportion of cases above the lower limit of a specified class interval. We can , develop the cumulative relative frequency distributions from the less than and greater than cumulative frequency distributions constructed earlier. By following the earlier example, we get:
| Class Interval (Years) | Cumulative Frequency (Cum. Freq.) | Cumulative Relative Frequency (Cum. Rel. Freq.) |
| Less than 25 | 5 | 5/30 or 16.7% |
| Less than 35 | 8 | 8/30 or 26.7% |
| Less than 45 | 15 | 15/30 or 50.0% |
| Less than 55 | 20 | 20/30 or 66.7% |
| Less than 65 | 23 | 23/30 or 76.7% |
| Less than 75 | 30 | 30/30 or 100% |
| Class Interval (Years) | Cumulative Frequency (Cum. Freq.) | Cumulative Relative Frequency (Cum. Rel. Freq.) |
| 15 and above 30 | 30 | 30/30 or 100% |
| 25 and above 35 | 25 | 25/30 or 83.3% |
| 35 and above 45 | 22 | 22/30 or 73.3% |
| 45 and above 55 | 15 | 15/30 or 50.0% |
| 55 and above 65 | 10 | 10/30 or 33.3% |
| 65 and above 75 | 7 | 7/30 or 23.3% |
In this example, 100 per cent of the persons are above 15 years of age, 73.3 per cent are above 35 years of age and so on. (It should be noted that the less than cumulative frequency distribution is summed up from top downwards and the greater than cumulative frequency distribution is summed from bottom upwards).