Question

Mode for the following distribution is \(17.5\) and \(x\) is less than 6 . Find \(x\). $$ \begin{array}{|l|c|c|c|c|c|} \hline \begin{array}{l} \text { Class } \\ \text { interval } \end{array} & 0-5 & 5-10 & 10-15 & 15-20 & 20-25 \\ \hline \text { Frequency } & 5 & 2 & 3 & 6 & \mathrm{x} \\ \hline \end{array} $$ (1) 3 (2) 2 (3) 4 (4) 5

Short answer

Answer: No, there is not enough information to uniquely determine the value of x.

Step-by-step solution

Determine the mode

Since the distribution's mode is 17.5, we know that the class interval with mode is 15-20. In this interval, the frequency is 6.

Check the given information

Since x is less than 6, it will not affect the mode as the frequency in the class interval 15-20 will still be the highest.

Choose the possible values of x from the options

Now let's check the given options for the value of x and see if they satisfy the conditions. (1) If x = 3, the frequency of the class interval 20-25 is 3 – less than the frequency of the class interval 15-20. This satisfies the condition, so it is a possible answer. (2) If x = 2, the frequency of the class interval 20-25 is 2 – less than the frequency of the class interval 15-20. This also satisfies the condition, so it is a possible answer too. (3) If x = 4, the frequency of the class interval 20-25 is 4 – less than the frequency of the class interval 15-20. This satisfies the condition, so it is another possible answer. (4) If x = 5, the frequency of the class interval 20-25 is 5 – less than the frequency of the class interval 15-20. This satisfies the condition, so it is a possible answer as well. Since all the possible values of x satisfy the given conditions and there are no more restrictions on the value of x, we can conclude that there is no unique answer. The problem does not provide enough information to uniquely determine the value of x.

Key concepts

Statistical Distributions

In the realm of statistics, understanding the fundamentals of how data is distributed is crucial for interpreting various forms of statistical analysis. A statistical distribution represents how the values of a variable are spread or dispersed. It gives us a holistic view of the dataset and provides insights into the probability of occurrence of different ranges of values. There are several types of distributions such as normal, binomial, and uniform, each with its unique properties and applications.

When we visualize data with a histogram or a frequency polygon, we're looking at the shape of the statistical distribution. The mode, the most frequently occurring value in a dataset, is a measure of central tendency that is intimately connected to the concept of statistical distributions. It gives us a quick glimpse at the peak of the distribution, where data points tend to cluster.

Understanding statistical distributions is imperative when dealing with complex data, as it aids in the selection of appropriate statistical tests and inferences based on the nature of the data, whether it is continuous, discrete, or categorical.

Frequency Distribution

A frequency distribution is a tabular or graphical representation that displays how often each different value in a set of data occurs. It is effectively an organized tally of a dataset. In the context of the problem provided, the table with class intervals and corresponding frequencies is a perfect example of a frequency distribution. Each entry or frequency in the table tells us how many times a value within a specific range, or class interval, appears in the dataset.

Importance of Frequency Distributions

Frequency distributions are important because they:

• Provide a snapshot of the data, making it easier to see patterns such as the mode.
• Help identify outliers or unusual values that might be worth further investigation.
• Allow the comparison between different sets of data on a unified scale.

Picturing the frequency distribution with a histogram, where each bar represents a class interval, can also provide a visual understanding of the mode. For instance, in the textbook exercise, the tallest bar would represent the mode of 17.5 within the class interval of 15-20.

Class Interval

The term class interval refers to a range of values within which data points are grouped in a frequency distribution table or histogram. In our example, class intervals such as 0-5, 5-10, and so on, define the ranges that are used to summarize the original data. Every class interval has an accompanying frequency which tells us how many data points fall within that range.

Choosing the right class interval is key to drawing meaningful conclusions from data. If intervals are too wide, important details might be lost. If they're too narrow, the overarching trends might become obscured by noise. A good class interval should strike a balance, providing a clear, yet detailed insight into the dataset's structure.

Understanding the concept of class intervals can significantly enhance one’s ability to interpret histograms and frequency distribution tables, which, in turn, facilitates a deeper comprehension of the data's underlying patterns and trends, including the mode.