Calculating the Mean of New Observations

The problem asks us to find the mean of three new observations added to a dataset, given the initial mean and the new mean after inclusion.

The mean of a set of observations is calculated by dividing the sum of all observations by the total number of observations.

The formula for the mean ($\bar{x}$) is:

$$\bar{x} = \frac{\text{Sum of observations}}{\text{Number of observations}}$$

From this, we can find the sum of observations if we know the mean and the number of observations:

$$\text{Sum of observations} = \bar{x} \times \text{Number of observations}$$

Step-by-Step Calculation

1. Calculate the Sum of the Initial 15 Observations

We are given that the mean of 15 observations is 14.

• Number of initial observations ($n_1$) = 15
• Mean of initial observations ($\bar{x}_1$) = 14

Sum of initial 15 observations ($S_1$) = $\bar{x}_1 \times n_1$

$$S_1 = 14 \times 15$$

$$S_1 = 210$$

So, the sum of the initial 15 observations is 210.

2. Calculate the Total Number of Observations

Three more observations are included in the dataset.

• Initial number of observations = 15
• Number of new observations = 3

Total number of observations ($n_2$) = Initial observations + New observations

$$n_2 = 15 + 3$$

$$n_2 = 18$$

The new total number of observations is 18.

3. Calculate the Sum of All 18 Observations

We are told that the new mean of the 18 observations becomes 13.

• New number of observations ($n_2$) = 18
• New mean ($\bar{x}_2$) = 13

Sum of all 18 observations ($S_2$) = $\bar{x}_2 \times n_2$

$$S_2 = 13 \times 18$$

To calculate $13 \times 18$:

• $13 \times 10 = 130$
• $13 \times 8 = 104$
• $130 + 104 = 234$

$$S_2 = 234$$

The sum of all 18 observations is 234.

4. Calculate the Sum of the Three New Observations

The sum of all 18 observations is the sum of the initial 15 observations plus the sum of the three new observations.

Sum of new observations ($S_{\text{new}}$) = Sum of all 18 observations - Sum of initial 15 observations

$$S_{\text{new}} = S_2 - S_1$$

$$S_{\text{new}} = 234 - 210$$

$$S_{\text{new}} = 24$$

The sum of the three new observations is 24.

5. Calculate the Mean of the Three New Observations

To find the mean of the three new observations, we divide their sum by the number of new observations (which is 3).

Number of new observations = 3

Sum of new observations ($S_{\text{new}}$) = 24

Mean of new observations ($\bar{x}_{\text{new}}$) = $\frac{S_{\text{new}}}{\text{Number of new observations}}$

$$\bar{x}_{\text{new}} = \frac{24}{3}$$

$$\bar{x}_{\text{new}} = 8$$

The mean of the three new observations is 8.

Let's summarise the steps and results:

The mean of the three new observations is 8.

Revision Table: Understanding Mean Calculations

Additional Information: Mean vs. Other Averages

The mean is one type of average, also known as the arithmetic mean. Other common averages include the median and the mode.

• Median: The middle value in a dataset that is ordered from least to greatest. If there's an even number of observations, it's the average of the two middle values. The median is less affected by extreme values (outliers) than the mean.
• Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.

The mean is widely used because it includes every value in the dataset in its calculation, but it can be skewed by very large or very small values.