Understanding Measures of Central Tendency and Dispersion

The question asks about how the mean and standard deviation of a dataset change when a constant value is added to each observation. This is a fundamental concept in statistics related to the properties of these measures.

Problem Description

We are given the following information for a set of 100 observations:

• Original Mean (\(\bar{x}\)) = 50
• Original Standard Deviation (\(\sigma\)) = 10
• Number of observations (n) = 100

We need to find the new mean and new standard deviation if 5 is added to each of the 100 observations.

Impact of Adding a Constant to Observations

Let \(x_1, x_2, \dots, x_{100}\) be the original 100 observations. The mean is given by \(\bar{x} = \frac{\sum x_i}{n}\). The standard deviation is given by \(\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\).

When a constant, say 'c', is added to each observation, the new observations become \(x'_i = x_i + c\). In this problem, the constant \(c = 5\). So, the new observations are \(x'_i = x_i + 5\).

Calculating the New Mean

The new mean, \(\bar{x}'\), is the sum of the new observations divided by the number of observations:

\(\bar{x}' = \frac{\sum x'_i}{n} = \frac{\sum (x_i + c)}{n}\)

We can split the sum:

\(\bar{x}' = \frac{\sum x_i + \sum c}{n}\)

The sum of a constant 'c' repeated 'n' times is \(n \times c\). So, \(\sum c = nc\).

\(\bar{x}' = \frac{\sum x_i + nc}{n} = \frac{\sum x_i}{n} + \frac{nc}{n}\)

We know that \(\frac{\sum x_i}{n}\) is the original mean \(\bar{x}\). So,

\(\bar{x}' = \bar{x} + c\)

This shows that if a constant 'c' is added to each observation, the new mean is the original mean plus 'c'.

Using the given values:

• Original Mean (\(\bar{x}\)) = 50
• Constant added (c) = 5
• New Mean (\(\bar{x}'\)) = 50 + 5 = 55

The new mean is 55.

Calculating the New Standard Deviation

The standard deviation measures the spread or dispersion of the data points around the mean. The formula for the new standard deviation, \(\sigma'\), based on the new observations \(x'_i\) and the new mean \(\bar{x}'\), is:

\(\sigma' = \sqrt{\frac{\sum (x'_i - \bar{x}')^2}{n}}\)

Substitute \(x'_i = x_i + c\) and \(\bar{x}' = \bar{x} + c\):

\(\sigma' = \sqrt{\frac{\sum ((x_i + c) - (\bar{x} + c))^2}{n}}\)

Simplify the term inside the parenthesis:

\((x_i + c) - (\bar{x} + c) = x_i + c - \bar{x} - c = x_i - \bar{x}\)

So, the formula for the new standard deviation becomes:

\(\sigma' = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\)

This is exactly the formula for the original standard deviation, \(\sigma\).

\(\sigma' = \sigma\)

This shows that if a constant 'c' is added to each observation, the standard deviation remains unchanged.

Using the given values:

• Original Standard Deviation (\(\sigma\)) = 10
• Constant added (c) = 5
• New Standard Deviation (\(\sigma'\)) = 10

The new standard deviation is 10.

Summary of Results

When 5 is added to each observation:

• New Mean = 55
• New Standard Deviation = 10

We need to find the option that shows the new mean and new standard deviation respectively.

Comparing with Options

Let's look at the options:

Option 3 matches our calculated new mean (55) and new standard deviation (10).

Revision Table: Properties of Mean and Standard Deviation

Additional Information: Measures of Central Tendency and Dispersion

• Mean: The average value of a dataset. It is sensitive to extreme values. Adding or subtracting a constant shifts the mean by that constant. Multiplying or dividing by a constant scales the mean by that constant.
• Standard Deviation: A measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the data points tend to be close to the mean (and each other), while a high standard deviation indicates that the data points are spread out over a wider range. Adding or subtracting a constant does not change the spread, only the location, so standard deviation is unaffected. Multiplying or dividing by a constant scales the spread proportionally.
• Other Measures:
  • Variance: The square of the standard deviation (\(\sigma^2\)). It is also a measure of dispersion. Adding a constant does not change variance. Multiplying by k changes variance by \(k^2\).
  • Median: The middle value of a dataset when ordered. Adding a constant adds the same constant to the median. Multiplying by a constant multiplies the median by that constant.
  • Mode: The value that appears most frequently. Adding a constant adds the same constant to the mode. Multiplying by a constant multiplies the mode by that constant.