Question

The value of the variable of the given data for which the number of observations with values less than it and grater than it are equal is (a) mean (b) median (c) mode (d) range

Short answer

(b) median

Step-by-step solution

Understand the concepts of mean, median, mode, and range

Mean: The mean of a dataset is the sum of all the values divided by the number of values. It gives the average of all the values in the data. Median: The median of a dataset is the middle value when the data is arranged in ascending or descending order. If there is an even number of values, then the median is the mean of the two middle values. The number of observations with values less than the median is equal to the number of observations with values greater than the median. Mode: The mode of a dataset is the value or values that occur most frequently in the data. There could be multiple modes, or none at all (a uniform distribution). Range: The range of a dataset is the difference between the maximum and minimum values in the data.

Identify the measure that satisfies the condition

In this case, the measure that represents the value of the variable for which the number of observations with values less than it and greater than it are equal is the median. So, the correct answer is: (b) median

Key concepts

Mean

The mean is a statistical measure that helps you find the average of a dataset. To calculate the mean, all the values in the set are added up and then divided by the number of observations. This measure is useful when you want a single value that represents the entire dataset in a compact form.
Understanding the mean is essential because it holds the ability to smooth out any small fluctuations within the data.
Here's how it works:

• Summarizes the data with a single number.
• Sensitive to outliers—high or low values can skew the mean.
• Utilizes every value within the data set.

For example, if five students scored 70, 80, 90, 85, and 75 on a test, the mean score would be calculated as follows:\[\text{Mean} = \frac{70 + 80 + 90 + 85 + 75}{5} = \frac{400}{5} = 80\]This result gives us a sense of the typical score among the students.

Mode

The mode is a unique statistical measure representing the most frequently occurring value(s) within a dataset. It indicates which number appears more often than others, which can be particularly useful when analyzing categorical data. Here’s what makes the mode standout:

• Helps identify the most common item or preference.
• A dataset can have one mode, more than one mode, or no mode at all.
• The only measure of central tendency that can be used with nominal data.

For instance, consider a small dataset representing shoe sizes: \( 7, 7, 8, 9, 9, 9, 10 \). In this case, the mode is 9, as it appears more often than any other number.
When you find the mode, you're essentially discovering the favorite or most popular item in your list!

Range

The range is another simple yet informative statistical measure, providing insight into the extent of variability in a dataset. It is calculated by finding the difference between the highest and lowest values. The range helps you understand the spread of the data.

• Shows the span or dispersion of data.
• Easy to calculate, but doesn’t consider all data points.
• Sensitive to outliers, where extreme values can affect the range significantly.

For a quick calculation, if you have a dataset \( 12, 18, 15, 10, 22 \), the range would be:\[\text{Range} = \text{Maximum} - \text{Minimum} = 22 - 10 = 12\]This tells us there is a spread of 12 units in the data. While it provides a basic glimpse into data variability, other measures such as variance or standard deviation can offer more detailed insights.

Statistical Measures

Statistical measures like mean, median, mode, and range are essential tools in data analysis. Each measure serves a different purpose, providing unique insights into the structure and characteristics of a dataset.
Here's why understanding these measures is crucial:

• Mean: Offers a comprehensive average by considering all data points.
• Median: Gives the middle value, ensuring half the data falls on either side.
• Mode: Identifies the most frequently occurring value.
• Range: Shows the total spread of the dataset.

These measures help you to summarize and understand data quickly. Knowing which measure to use can better equip you to analyze and interpret statistical information accurately.
When applied correctly, statistical measures allow you to draw meaningful conclusions, predict behaviors, and make informed decisions based on data.