Finding the Correct Mean of Student Scores

This problem involves calculating the correct mean of a set of student scores after identifying errors in the initial data entry. The mean is a fundamental statistical measure that represents the average value of a dataset. When data points are recorded incorrectly, the calculated mean will also be incorrect. To find the correct mean, we must first determine the correct sum of the scores and then divide by the total number of students.

Understanding the Concept of Mean

The mean (or average) of a dataset is calculated by summing all the values in the dataset and dividing by the number of values. The formula for the mean ($\bar{x}$) is:

$\bar{x} = \frac{\sum x}{n}$

Where:

• $\sum x$ is the sum of all the values
• $n$ is the number of values

In this problem, the 'values' are the scores obtained by the students.

Calculating the Incorrect Sum of Scores

We are given the following information:

• Number of students ($n$) = 50
• Incorrect mean = 79.5

The incorrect sum of scores can be found by rearranging the mean formula:

Incorrect Sum = Incorrect Mean $\times$ Number of students

Incorrect Sum = $79.5 \times 50$

Let's calculate this:

Incorrect Sum = $79.5 \times 50 = 3975$

So, the sum of the scores as initially calculated with the errors was 3975.

Adjusting the Sum for Incorrect Readings

We are told that two scores were read incorrectly:

• One score was read as 94 instead of 49.
• Another score was read as 69 instead of 89.

To find the correct sum, we need to remove the incorrect scores from the incorrect sum and add the correct scores.

Incorrect scores that were included in the sum: 94 and 69.

Correct scores that should have been included: 49 and 89.

Total value of incorrect scores included = $94 + 69 = 163$

Total value of correct scores that should be included = $49 + 89 = 138$

The adjustment needed for the sum is: (Sum of correct scores) - (Sum of incorrect scores)

Adjustment = $138 - 163 = -25$

Now, we can find the correct sum of scores:

Correct Sum = Incorrect Sum + Adjustment

Correct Sum = $3975 + (-25)$

Correct Sum = $3975 - 25 = 3950$

The correct sum of the scores is 3950.

Calculating the Correct Mean

Now that we have the correct sum of scores and the number of students, we can calculate the correct mean using the mean formula:

Correct Mean = $\frac{\text{Correct Sum}}{\text{Number of students}}$

Correct Mean = $\frac{3950}{50}$

Let's calculate this:

Correct Mean = $\frac{3950}{50} = \frac{395}{5}$

Correct Mean = 79

The correct mean of the scores obtained by the 50 students is 79.

Revision Table: Key Statistical Concepts

Additional Information: Dealing with Data Errors

Errors in data collection or entry are common in statistics. Recognizing and correcting these errors is crucial for obtaining accurate results and making reliable conclusions. Common types of errors include:

• Transcription Errors: Mistakes made when copying data from one source to another, as seen in this problem where scores were written down incorrectly.
• Measurement Errors: Inaccuracies in the tools or methods used to measure data.
• Sampling Errors: Differences between a sample and the population it represents, which are not necessarily "errors" in measurement but affect representativeness.

When dealing with incorrect entries in sum-based statistics like the mean, the method used in this problem (subtracting the wrong values and adding the correct values to the total sum) is the standard approach. For other statistics like variance or standard deviation, correcting the data and recalculating is necessary.

Ensuring data accuracy is a critical step before performing any statistical analysis to ensure the validity of the findings.