Question

Let mean of 100 data points be 55 and variance be \(16 .\) Now if every data point is increased by 2 units, then the sum of mean and standard deviation of new data points will be (1) 71 (2) 59 (3) 63 (4) 61 (5) 55

Short answer

61

Step-by-step solution

Understand the given data

Initially, there are 100 data points with a mean \( \mu = 55 \) and variance \( \sigma^2 = 16 \).

Calculate initial standard deviation

The initial standard deviation is the square root of the variance: \[ \sigma = \sqrt{16} = 4 \]

Adjust the mean for new data

If each data point is increased by 2 units, the new mean will be the old mean plus 2. \[ \mu_{new} = 55 + 2 = 57 \]

Determine the new standard deviation

Adding a constant to each data point does not change the standard deviation. Thus, the new standard deviation is still \( \sigma_{new} = 4 \).

Calculate the sum of the new mean and new standard deviation

Add the new mean and the new standard deviation: \[ 57 + 4 = 61 \]

Key concepts

mean and variance

In statistics, the mean and variance are fundamental concepts used to describe data sets.
The mean, often referred to as the average, is calculated by summing up all the data points and then dividing by the number of points.
It's a measure of central tendency, giving us an idea of the 'central' value of the data set.
Variance, on the other hand, measures the spread of the data points around the mean.
Specifically, it is the average of the squared differences from the mean.
Variance helps us understand how much the data points differ from the mean.
In our example, the mean of 100 data points is 55, and the variance is 16.
This means, on average, each data point is 16 units squared away from the mean.

standard deviation

Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean.
It is the square root of the variance, providing a measure that is in the same unit as the data, making it easier to interpret.
In mathematical terms, the standard deviation \( \sigma \) is given by:
$$ \sigma = \sqrt{\sigma^2} $$
For the dataset with variance 16, the standard deviation is:
\[ \sigma = \sqrt{16} = 4 \]
Standard deviation gives us a clear idea of how spread out the data points are.
A smaller standard deviation indicates that the data points are close to the mean, whereas a larger standard deviation suggests higher variability.

data transformation

Data transformation refers to the process of changing the values of data points in a dataset.
One common transformation is adding or subtracting a constant value to each data point.
In our example, each data point was increased by 2 units.
When we increase each data point by a constant, the mean will also increase by the same amount.
In our case, the new mean becomes 55 + 2 = 57.
However, adding a constant does not affect the spread of the data points.
Thus, the variance and standard deviation remain unchanged.
This is why the new standard deviation remains 4, the same as the initial standard deviation.