The mean of 12 observations is 15. One more observation is included and the new mean becomes 16. The 13th observation is
28
This problem involves calculating the sum of observations given the mean, and then using the change in mean after adding a new observation to find the value of that new observation.
Understanding the Concept of Mean
The mean (or average) of a set of observations is calculated by dividing the sum of all observations by the total number of observations. The formula is:
$$\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}$$
From this, we can also find the sum of observations if we know the mean and the number of observations:
$$\text{Sum of observations} = \text{Mean} \times \text{Number of observations}$$
Calculating the Initial Sum of Observations
We are given that the mean of 12 observations is 15. Using the formula for the sum of observations:
Number of observations initially = 12
Initial mean = 15
Initial sum of 12 observations = Initial mean $\times$ Number of observations
Initial sum = $15 \times 12$
Initial sum = 180
Calculating the New Sum of Observations
One more observation is included, making the total number of observations 13. The new mean is given as 16.
Number of observations finally = 12 + 1 = 13
New mean = 16
New sum of 13 observations = New mean $\times$ Number of observations
New sum = $16 \times 13$
New sum = 208
Finding the 13th Observation
The new sum of 13 observations includes the sum of the original 12 observations plus the value of the 13th observation. Therefore, the value of the 13th observation is the difference between the new sum and the initial sum.
Value of 13th observation = New sum of 13 observations - Initial sum of 12 observations
Value of 13th observation = $208 - 180$
Value of 13th observation = 28
So, the 13th observation is 28.
Summary of Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Initial Number of Observations | Given | 12 |
| 2 | Initial Mean | Given | 15 |
| 3 | Initial Sum of 12 Observations | Mean $\times$ Count | $15 \times 12 = 180$ |
| 4 | New Number of Observations | Initial Count + 1 | $12 + 1 = 13$ |
| 5 | New Mean | Given | 16 |
| 6 | New Sum of 13 Observations | New Mean $\times$ New Count | $16 \times 13 = 208$ |
| 7 | Value of 13th Observation | New Sum - Initial Sum | $208 - 180 = 28$ |
Revision Table: Mean and Sum Concepts
| Concept | Definition/Formula | Notes |
|---|---|---|
| Mean ($\bar{x}$) | $\bar{x} = \frac{\sum x}{n}$ (Sum of observations / Number of observations) | Measure of central tendency |
| Sum of Observations ($\sum x$) | $\sum x = \bar{x} \times n$ (Mean $\times$ Number of observations) | Useful for finding a missing observation |
| Effect of New Observation | New Sum = Old Sum + New Observation | Allows finding the new observation value |
Additional Information: Mean Calculation and Applications
The mean is a fundamental concept in statistics. It gives us a single value that represents the center of a dataset. While easy to calculate, it can be affected by extreme values (outliers).
- Calculating the mean helps in summarizing data.
- Comparing means of different groups allows for analysis.
- The concept of mean is used in many statistical calculations, including variance and standard deviation.
- Understanding how adding or removing data points affects the mean is important for data analysis.
This problem demonstrates a common application of the mean formula: using known means to calculate sums and infer the value of an unknown data point.
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?
