Question

The mean of 16 observations is 16 . And if one observation 16 is deleted and three observations 5,5 and 6 are included, then find the mean of the final observations. (1) 16 (2) \(15.5\) (3) \(13.5\) (4) None of these

Short answer

Answer: (4) None of these

Step-by-step solution

Find the sum of the original observations

Since the mean of the original 16 observations is 16, we can find the sum of the original observations by multiplying the mean by the number of observations: Sum of original observations = mean × number of observations = 16 × 16 = 256

Find the sum of the adjusted observations

First, remove the observation with a value of 16 from the sum: Adjusted sum = original sum - removed observation = 256 - 16 = 240 Now, add the three new observations (5, 5, and 6) to the adjusted sum: Adjusted sum = 240 + 5 + 5 + 6 = 256

Find the mean of the adjusted observations

Now that we have the sum of the adjusted observations, we can calculate the mean of the final observations. Since we removed one observation and added three, we now have 18 total observations: Mean of adjusted observations = adjusted sum / number of adjusted observations = 256 / 18 = 14.22 Since 14.22 is not one of the given options, the correct answer is: (4) None of these

Key concepts

Statistics Problem Solving

When faced with statistics problems, the key is breaking them down into manageable steps. It's important to start by understanding what the problem is asking. Before diving into calculations, identify the main concepts. For the exercise given, you needed to find the mean of a data set after making changes to it.

Typically, a problem-solving strategy includes:

• Identifying what information is provided and what is required.
• Breaking the problem into smaller chunks that you can solve sequentially.
• Carefully applying relevant statistical formulas.

In this exercise, the problem was separated into steps like calculating initial totals, adjusting these totals, then finding a new mean. Following a step-by-step strategy ensures accuracy and clarity in statistical problem solving.

Addition and Subtraction in Statistics

Addition and subtraction are fundamental in statistics, especially when dealing with data sets and finding measures like the mean. In the provided exercise, we used these operations to alter the data set.

First, we subtracted an observation (16 in this case) from the total sum of original observations. Then, we added three new observations: 5, 5, and 6. This adjusted the total sum and directly influenced the calculation of the mean.

Using these operations effectively allows us to manipulate and update data sets as needed. Always remember:

• Subtraction helps you remove unwanted or erroneous data entries.
• Addition helps you update the data set with new or corrected data.

Mastering these operations within the context of statistics will increase your ability to handle complex data sets smoothly.

Observations in Data Sets

An observation in a data set is essentially an individual data item or value. Understanding how changes in observations affect your data is crucial. In this problem, we started with 16 observations, each contributing to the overall mean.

Removing and adding observations changes the dataset and impacts calculations. Here, by deleting one observation with a value of 16 and incorporating three new observations (5, 5, and 6), we modified the number and sum of observations. Now, we deal with 18 observations.

When handling observations, consider:

• Each observation affects the sum and the resulting mean.
• Adding more observations usually increases the complexity of calculations but provides richer data insights.

Properly managing these observations is key to maintaining accuracy in statistical analyses.

Arithmetic Mean

The arithmetic mean, often called the average, is a measure of central tendency. It gives a single value that represents the center of a data set. It's calculated by dividing the sum of all observations by the number of observations.

In our problem, the original mean was straightforwardly calculated using the given mean and the number of observations. When the observations were altered, we recomputed the mean by dividing the new sum (256) by the new number of observations (18). The result was 14.22, which turned out to be none of the provided options.

When calculating the arithmetic mean, always remember:

• Ensure all observations are included in the sum.
• Adjust the number of observations when modifying the data set.
• An inaccurate count or sum can yield incorrect results.

Understanding the arithmetic mean is essential, as it forms the basis for more complex statistical measures and interpretations.