Understanding the Average and the Effect of Subtraction

This problem involves calculating the value of a number 'M' that is subtracted from each number in a set, resulting in a change in the average of the set.

The average of a set of numbers is calculated by summing all the numbers and dividing by the count of numbers.

\text{Average} = \frac{\text{Sum of Numbers}}{\text{Number of Numbers}}

Analyzing the Given Information

• Initial number of values: 24
• Initial average: 43
• Action: A number 'M' is subtracted from each of the 24 numbers.
• New average: 38

Solving the Problem Using the Property of Averages

A key property of averages states that if a constant value is added to or subtracted from every number in a set, the average of the new set will be the original average plus or minus that constant value, respectively.

In this case, 'M' is subtracted from each number. Therefore, the new average is equal to the original average minus 'M'.

\text{New Average} = \text{Original Average} - \text{M}

We are given the original average (43) and the new average (38). We can substitute these values into the equation:

$$38 = 43 - M$$

Now, we need to solve for M. We can rearrange the equation to isolate M:

$$M = 43 - 38$$

Performing the subtraction:

$$M = 5$$

So, the value of M is 5.

Alternative Approach: Using the Sum

We can also solve this by calculating the sum of the numbers.

Step 1: Calculate the initial sum.

Using the formula: Sum = Average \(\times\) Number of Numbers

$$ \text{Initial Sum} = 43 \times 24 $$

$$ \text{Initial Sum} = 1032 $$

Step 2: Calculate the new sum.

The new average is 38, and there are still 24 numbers.

$$ \text{New Sum} = 38 \times 24 $$

$$ \text{New Sum} = 912 $$

Step 3: Relate the sums to M.

When M is subtracted from each of the 24 numbers, the total sum decreases by \(24 \times M\).

$$ \text{New Sum} = \text{Initial Sum} - (24 \times M) $$

Substitute the calculated sums:

$$ 912 = 1032 - (24 \times M) $$

Rearrange the equation to solve for \(24 \times M\):

$$ 24 \times M = 1032 - 912 $$

$$ 24 \times M = 120 $$

Now, solve for M:

$$ M = \frac{120}{24} $$

$$ M = 5 $$

Both methods confirm that the value of M is 5.

Revision Table: Key Concepts in Average

Additional Information: Understanding Changes in Data Sets

Understanding how operations on individual data points affect summary statistics like the average is crucial in data analysis and statistics.

• Consistent Operations: Adding or subtracting a constant from every number in a set shifts the entire data set uniformly along the number line. This uniform shift directly impacts the average by the same amount.
• Non-Consistent Operations: If different values are added or subtracted from different numbers in the set, the effect on the average is the total change in sum divided by the number of values.
• Impact on Other Statistics: While adding/subtracting a constant affects the mean (average), it does not change measures of dispersion like the range, variance, or standard deviation. Multiplying/dividing by a constant, however, affects both the mean and the dispersion measures.

This problem specifically used the property related to subtracting a constant, which simplifies the calculation significantly.