Understanding the Arithmetic Mean Problem
This question asks us to find the new arithmetic mean of a set of observations after applying a specific change: multiplying every observation by 5. We are given the original mean and the number of observations.
Given Information
- Number of observations (\(N\)): 200
- Original arithmetic mean (\(\bar{X}\)): 60
- Operation applied to each observation: Multiplication by 5
Arithmetic Mean Basics
The arithmetic mean, often called the average, is calculated by summing up all the values in a dataset and then dividing by the count of those values. The formula is:
\(\bar{X} = \frac{\sum_{i=1}^{N} x_i}{N}\)
Where '\(\sum_{i=1}^{N} x_i\)' represents the sum of all observations (\(x_1, x_2, \dots, x_N\)) and \(N\) is the total number of observations.
Step-by-Step Calculation
First, let's use the given information to find the sum of the original observations.
We know that \(\bar{X} = 60\) and \(N = 200\). Plugging these into the formula:
\(60 = \frac{\sum_{i=1}^{200} x_i}{200}\)
To find the sum of the original observations, we multiply the mean by the number of observations:
\(\text{Sum of original observations} = \sum_{i=1}^{200} x_i = 60 \times 200 = 12000\)
Next, consider the effect of multiplying each observation by 5. Let the new observations be \(x'_i\). Then, \(x'_i = 5 \times x_i\) for each \(i\) from 1 to 200.
The sum of the new observations is:
\(\text{Sum of new observations} = \sum_{i=1}^{200} x'_i = \sum_{i=1}^{200} (5 x_i)\)
Using the properties of summation, we can pull the constant factor 5 out:
\(\sum_{i=1}^{200} (5 x_i) = 5 \times \sum_{i=1}^{200} x_i\)
We already calculated the sum of the original observations (\(12000\)), so:
\(\text{Sum of new observations} = 5 \times 12000 = 60000\)
Finally, to find the new arithmetic mean (\(\bar{X}_{new}\)), we divide the sum of the new observations by the number of observations (which is still 200):
\(\bar{X}_{new} = \frac{\text{Sum of new observations}}{N} = \frac{60000}{200}\)
\(\bar{X}_{new} = 300\)
Using Properties of Mean for a Quicker Solution
There's a useful property of the arithmetic mean that simplifies this type of problem significantly. The property states:
If each observation in a dataset is multiplied by a constant value \(k\), the mean of the dataset is also multiplied by the same constant \(k\).
Mathematically, if \(x'_i = k \times x_i\), then the new mean \(\bar{X}_{new}\) is:
\(\bar{X}_{new} = k \times \bar{X}\)
Applying this property to our problem:
- Original Mean (\(\bar{X}\)) = 60
- Constant Multiplier (\(k\)) = 5
New Mean (\(\bar{X}_{new}\)):
\(\bar{X}_{new} = 5 \times 60\)
\(\bar{X}_{new} = 300\)
Both methods confirm that the new arithmetic mean is 300.
Similar Questions
What is the median of the distribution ?
- What is the mode of the distribution?
- Consider the following data :
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct? - Which measure of central tendency is least affected by the presence of extreme observations in the data?
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- What is the value of \(u\)?
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Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
