Understanding the Mean of Observations

The mean (or average) of a set of observations is calculated by dividing the sum of all observations by the total number of observations. This fundamental concept is key to solving problems involving changes in a dataset's mean, such as when observations are added or removed.

In this problem, we are given the mean of an initial set of observations. Then, some observations are removed, and the mean of the remaining observations is provided. We need to find the mean of the observations that were removed (discarded).

Step-by-Step Calculation of Discarded Mean

Let's break down the problem into smaller steps to calculate the mean of the discarded observations.

1. Calculate the Sum of the Initial Observations

We are given:

• Number of initial observations: \(N_1 = 12\)
• Mean of initial observations: \(\text{Mean}_1 = 75\)

The formula for the mean is:

\(\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}\)

Rearranging the formula to find the sum:

\(\text{Sum} = \text{Mean} \times \text{Number of observations}\)

So, the sum of the initial 12 observations is:

\(\text{Sum}_1 = \text{Mean}_1 \times N_1 = 75 \times 12\)

Calculating the sum:

\(\text{Sum}_1 = 900\)

2. Calculate the Sum of the Remaining Observations

Two observations are discarded, so the number of remaining observations is:

Number of remaining observations: \(N_2 = 12 - 2 = 10\)

We are given the mean of these remaining observations:

Mean of remaining observations: \(\text{Mean}_2 = 65\)

Using the same formula for the sum:

\(\text{Sum}_2 = \text{Mean}_2 \times N_2 = 65 \times 10\)

Calculating the sum:

\(\text{Sum}_2 = 650\)

3. Calculate the Sum of the Discarded Observations

The initial sum of observations is the sum of the remaining observations plus the sum of the discarded observations.

Sum of initial observations = Sum of remaining observations + Sum of discarded observations

\(\text{Sum}_1 = \text{Sum}_2 + \text{Sum}_{\text{discarded}}\)

To find the sum of the discarded observations, we subtract the sum of the remaining observations from the initial sum:

\(\text{Sum}_{\text{discarded}} = \text{Sum}_1 - \text{Sum}_2 = 900 - 650\)

Calculating the sum of discarded observations:

\(\text{Sum}_{\text{discarded}} = 250\)

4. Calculate the Mean of the Discarded Observations

We know that 2 observations were discarded.

Number of discarded observations: \(N_{\text{discarded}} = 2\)

We have calculated the sum of these 2 discarded observations:

Sum of discarded observations: \(\text{Sum}_{\text{discarded}} = 250\)

Now, we can find the mean of the discarded observations using the mean formula:

\(\text{Mean}_{\text{discarded}} = \frac{\text{Sum}_{\text{discarded}}}{\text{Number}_{\text{discarded}}} = \frac{250}{2}\)

Calculating the mean of the discarded observations:

\(\text{Mean}_{\text{discarded}} = 125\)

Therefore, the mean of the two discarded observations is 125.

Summary of Calculations

Mean of discarded observations = \(\frac{\text{Sum of Discarded Observations}}{\text{Number of Discarded Observations}} = \frac{250}{2} = 125\).

Revision Table: Key Concepts

Additional Information on Mean and Data Changes

The mean is a simple and widely used measure of central tendency. However, it is sensitive to outliers (extremely large or small values). When observations are added or removed, the sum of the observations changes, which in turn affects the mean.

• Adding an observation: If you add an observation, the sum increases. If the added observation is greater than the original mean, the new mean will be higher. If it's less, the new mean will be lower.
• Removing an observation: When observations are removed, the sum decreases. If the removed observations had a mean greater than the original mean, the new mean (of the remaining data) will be lower. If they had a mean less than the original mean, the new mean will be higher. In this problem, the mean decreased from 75 to 65, indicating the discarded observations had a mean greater than 65 (and indeed, their mean was 125, much higher than 75).

Problems like this highlight the relationship between the mean, the sum of observations, and the number of observations. Understanding this relationship allows you to work backward or forward when parts of the data are known or changed.