Problem 45
Question
If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately (A) \(22.0\) (B) \(20.5\) (C) \(25.5\) (D) \(24.0\)
Step-by-Step Solution
Verified Answer
The mode is approximately 24, which corresponds to option (D).
1Step 1: Understand the Relationship between Mean, Median, and Mode
In a frequency distribution, when it is skewed, the relationship between the mean, median, and mode is given by the empirical formula: \( \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \). This formula helps us estimate the mode when the distribution is skewed.
2Step 2: Substitute Given Values into the Formula
Substitute the given values from the problem into the empirical formula. Here, the mean is 21 and the median is 22. The formula becomes: \( \text{Mode} = 3 \times 22 - 2 \times 21 \).
3Step 3: Calculate the Mode
Now, calculate the mode using the substituted values. First calculate \(3 \times 22\), which equals 66. Then calculate \(2 \times 21\), which equals 42. Subtracting these gives \(66 - 42 = 24\).
4Step 4: Select the Closest Answer from Options
The calculated mode is 24. This matches option (D) in the given multiple-choice answers.
Key Concepts
Frequency DistributionMean, Median, and Mode RelationshipEmpirical Formula for Skewness
Frequency Distribution
Frequency distribution is a crucial concept in statistics. It refers to a summary that shows how frequently each different value in a set of data occurs. By creating a frequency distribution, it becomes easier to analyze and understand the shape of the data set.
Imagine you have a collection of test scores from a class of students. Instead of just listing all the scores, a frequency distribution helps to cluster these scores into categories or intervals. By doing so, you get a clear idea of which scores appear most frequently.
For example:
The importance of frequency distribution lies in its ability to make complex data more comprehensible. It highlights patterns and facilitates the identification of central values and spread, leading to insights about the data's characteristics.
Imagine you have a collection of test scores from a class of students. Instead of just listing all the scores, a frequency distribution helps to cluster these scores into categories or intervals. By doing so, you get a clear idea of which scores appear most frequently.
For example:
- 40-49: 5 students
- 50-59: 8 students
- 60-69: 12 students
- 70-79: 10 students
The importance of frequency distribution lies in its ability to make complex data more comprehensible. It highlights patterns and facilitates the identification of central values and spread, leading to insights about the data's characteristics.
Mean, Median, and Mode Relationship
In statistics, the mean, median, and mode are measures of central tendency. They summarize a set of data into a single representative value, aiding in data interpretation.
The **mean** is the average of all data points. It's calculated by summing all values and dividing by the total number of values. The **median** is the middle value when data is ordered from smallest to largest, and the **mode** is the value that appears most frequently in the dataset.
These values help understand the distribution of data. They also have a special relationship, especially in skewed distributions. The **empirical formula** for this is:
When used together, the mean, median, and mode provide a comprehensive insight into the characteristics and shape of the data.
The **mean** is the average of all data points. It's calculated by summing all values and dividing by the total number of values. The **median** is the middle value when data is ordered from smallest to largest, and the **mode** is the value that appears most frequently in the dataset.
These values help understand the distribution of data. They also have a special relationship, especially in skewed distributions. The **empirical formula** for this is:
- Mode = 3 × Median - 2 × Mean
When used together, the mean, median, and mode provide a comprehensive insight into the characteristics and shape of the data.
Empirical Formula for Skewness
Skewness in a distribution refers to the asymmetry in the data. A distribution can be positively skewed (right-skewed) or negatively skewed (left-skewed). The empirical formula involving the mean, median, and mode becomes particularly necessary to understand these skews.
In practice, when the mean, median, and mode do not coincide, it indicates a skewed distribution. The empirical formula for estimating the mode helps in quantifying this skewness:
This relationship is helpful to infer whether the tail of the distribution is skewed positively or negatively. A numerical mode different from mean and median suggests a significant skew, providing insights into the nature of the data's asymmetry. By understanding and applying these concepts, statisticians can better interpret and represent real-world data distributions.
In practice, when the mean, median, and mode do not coincide, it indicates a skewed distribution. The empirical formula for estimating the mode helps in quantifying this skewness:
- Mode = 3 × Median - 2 × Mean
This relationship is helpful to infer whether the tail of the distribution is skewed positively or negatively. A numerical mode different from mean and median suggests a significant skew, providing insights into the nature of the data's asymmetry. By understanding and applying these concepts, statisticians can better interpret and represent real-world data distributions.
Other exercises in this chapter
Problem 43
Consider the following statements (A) Mode can be computed from histogram (B) Median is not independent of change of scale (C) Variance is independent of change
View solution Problem 44
In a series of \(2 n\) observations, half of them equal \(a\) and remaining half equal \(-a\). If the standard deviation of the observations is 2 , then \(|a|\)
View solution Problem 46
Let \(x_{1}, x 2, \ldots, x_{n}\) be \(n\) observations such that \(\sum x_{i}^{2}=400\) and \(\sum x_{i}=80 .\) Then a possible value of \(n\) among the follow
View solution Problem 47
Suppose a population \(A\) has 100 observations 101 , \(102, \ldots, 200\), and another population \(B\) has 100 observations \(151,152, \ldots, 250\). If \(V_{
View solution