The mean of 15 observations is 14. Three more observations are included and the new mean becomes 13. The mean of three new observations is: A. 6 B. 7 C. 8 D. 11
C
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:
| Description | Value | Calculation |
|---|---|---|
| Initial Number of Observations ($n_1$) | 15 | Given |
| Initial Mean ($\bar{x}_1$) | 14 | Given |
| Sum of Initial Observations ($S_1$) | 210 | $14 \times 15$ |
| Number of New Observations | 3 | Given |
| Total Number of Observations ($n_2$) | 18 | $15 + 3$ |
| New Mean ($\bar{x}_2$) | 13 | Given |
| Sum of All Observations ($S_2$) | 234 | $13 \times 18$ |
| Sum of New Observations ($S_{\text{new}}$) | 24 | $234 - 210$ |
| Mean of New Observations ($\bar{x}_{\text{new}}$) | 8 | $24 / 3$ |
The mean of the three new observations is 8.
Revision Table: Understanding Mean Calculations
| Concept | Formula | Explanation |
|---|---|---|
| Mean ($\bar{x}$) | $\frac{\Sigma x}{n}$ | Sum of observations divided by the number of observations. |
| Sum of Observations ($\Sigma x$) | $\bar{x} \times n$ | Mean multiplied by the number of observations. Useful for finding total value. |
| Effect of Adding Observations | Change in Sum / Change in Number | Adding new observations changes both the total sum and the total count, affecting the mean. |
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.
Important Questions from Elementary Statistics
Demand for seats in a university is at its highest in the fall; demand also trends to grow and fall off in 25 year waves. In time service forecasting, the former demand characteristic would be called ______ and the latter would be called _______.
The system of combining two or more overlapping series of index numbers to obtain a single continuous series is called
The rise in the number of patients due to heatstroke is an example of:
According to government data, 24 percent of teenagers in India under the age of 18 years live in households with incomes that are classified at a particular income level. A simple random sample of 400 teenagers in India under the age of 18 years was selected for a study of learning. If the government data is correct, which of the following best approximates the probability that at least 27 per cent of the teenagers in the sample live in households that are classified at a particular income level?
Which index satisfies the factor reversal test?
