The arithmetic mean of 100 observation is 40. Later, it was found that an observation '53' was wrongly read as '83'. What is the correct arithmetic mean?
39.7
Understanding Arithmetic Mean and Error Correction
The arithmetic mean, often simply called the mean, is a measure of central tendency. It is calculated by summing all the observations in a dataset and dividing by the total number of observations. The formula for the arithmetic mean (\(\bar{x}\)) is:
\[ \bar{x} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]In this problem, we are given an initial arithmetic mean that is incorrect because one observation was misread. We need to find the correct arithmetic mean.
Calculating the Original (Incorrect) Sum
We are given the following information:
- Number of observations (n) = 100
- Incorrect arithmetic mean (\(\bar{x}_{initial}\)) = 40
Using the formula for the mean, we can find the total sum of the observations based on the initial, incorrect calculation:
\[ \text{Sum of observations} = \text{Arithmetic Mean} \times \text{Number of observations} \] \[ \Sigma x_{initial} = \bar{x}_{initial} \times n \] \[ \Sigma x_{initial} = 40 \times 100 \] \[ \Sigma x_{initial} = 4000 \]So, the sum of the 100 observations, as initially calculated, was 4000.
Correcting the Sum of Observations
The problem states that an observation '53' was wrongly read as '83'. This means the value 83 was included in the sum (4000) instead of the correct value 53.
To find the correct sum of observations (\(\Sigma x_{correct}\)), we need to remove the incorrect value (83) from the initial sum and add the correct value (53):
\[ \Sigma x_{correct} = \Sigma x_{initial} - \text{Incorrect value} + \text{Correct value} \] \[ \Sigma x_{correct} = 4000 - 83 + 53 \] \[ \Sigma x_{correct} = 4000 - 30 \] \[ \Sigma x_{correct} = 3970 \]The correct sum of the 100 observations is 3970.
Calculating the Correct Arithmetic Mean
Now that we have the correct sum of observations and the number of observations (which remains 100), we can calculate the correct arithmetic mean (\(\bar{x}_{correct}\)):
\[ \bar{x}_{correct} = \frac{\text{Correct Sum of observations}}{\text{Number of observations}} \] \[ \bar{x}_{correct} = \frac{\Sigma x_{correct}}{n} \] \[ \bar{x}_{correct} = \frac{3970}{100} \] \[ \bar{x}_{correct} = 39.7 \]Therefore, the correct arithmetic mean is 39.7.
Summary of Steps for Correcting Mean
Here is a summary of the process followed:
- Calculated the initial sum using the given incorrect mean and number of observations.
- Identified the incorrect observation included and the correct observation that should have been included.
- Adjusted the sum by subtracting the incorrect value and adding the correct value.
- Calculated the correct mean using the corrected sum and the total number of observations.
| Item | Value | Calculation/Notes |
|---|---|---|
| Number of observations (n) | 100 | Given |
| Initial Mean (\(\bar{x}_{initial}\)) | 40 | Given (incorrect) |
| Initial Sum (\(\Sigma x_{initial}\)) | 4000 | \(40 \times 100\) |
| Incorrectly read value | 83 | Should be removed |
| Correct value | 53 | Should be added |
| Correction factor | -30 | \(53 - 83\) |
| Corrected Sum (\(\Sigma x_{correct}\)) | 3970 | \(4000 - 30\) |
| Correct Mean (\(\bar{x}_{correct}\)) | 39.7 | \(3970 / 100\) |
Revision Table: Mean Calculation with Errors
Understanding how to correct the mean when errors in data entry occur is important. The method involves adjusting the total sum of observations.
Common types of errors in mean calculation include:
- An observation is wrongly copied.
- An observation is omitted.
- An observation is included twice.
In this case, it was a wrong copying error (53 read as 83).
Additional Information: Properties of Arithmetic Mean
The arithmetic mean has several important properties:
- The sum of the deviations of the observations from their arithmetic mean is always zero. \(\Sigma (x_i - \bar{x}) = 0\).
- If each observation in a dataset is increased or decreased by a constant value 'c', the mean also increases or decreases by 'c'.
- If each observation is multiplied or divided by a constant value 'c', the mean also gets multiplied or divided by 'c'.
- The mean is affected by extreme values (outliers).
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