Question

In a series of observations, S.D. is 7 and mean is 28 . find the coefficient of variation. (1) 4 (2) \(1 / 4\) (3) 25 (4) \(12.5\)

Short answer

Answer: 25%

Step-by-step solution

Identify the Mean and Standard Deviation

The given mean is 28, and the standard deviation is 7.

Use the Formula for Coefficient of Variation

The formula for the coefficient of variation is: CV = (SD/Mean) × 100.

Plug in the Given Values

Using the given values, we have CV = (7/28) × 100.

Simplify the Expression

CV = (1/4) × 100 = 0.25 × 100 = 25%. So, the correct answer is (3) 25.

Key concepts

Standard Deviation

Standard deviation is a vital concept in statistics. It measures the amount of variation or dispersion within a set of data points. A lower standard deviation indicates that data points are generally close to the mean, whereas a higher standard deviation shows data points spread out over a broader range of values. This is useful in assessing the consistency and reliability of data.

To calculate the standard deviation, every observation's deviation from the mean is squared, then averaged, and finally, the square root of that average is taken to return to the original unit of measure. The formula for standard deviation is:

• \[ SD = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N}} \]

where \( x_i \) represents each data point, \( \bar{x} \) is the mean of the data, and \( N \) is the number of observations.

Mean in Statistics

The mean, also known as the average, is a measure of the central tendency of a dataset. It gives you a central value around which all other observations are distributed. To find the mean, sum up all the dataset's values and then divide by the number of observations.

The formula for calculating the mean is quite straightforward:

• \[ \bar{x} = \frac{\sum x_i}{N} \]

Here, \( \sum x_i \) refers to the summation of all data points, and \( N \) is the total number of observations. The mean is particularly useful in determining the overall trend of data, although it may not always reflect real-world scenarios when outliers are present.

Mathematical Formula Application

Mathematical formulas simplify complex relationships into understandable calculations. Applying the right formula can help solve statistical problems efficiently and with accuracy.

In the given exercise, the focus was on the Coefficient of Variation (CV), which measures relative variability:

• The Coefficient of Variation is helpful for comparing the degree of variation from one dataset to another, even if the means are drastically different.
• The formula used here is:\[CV = \left(\frac{SD}{Mean}\right) \times 100.\]By plugging in the standard deviation (7) and mean (28), the solution shows that the CV is 25%.

This means per 100 units of the mean, there is a variation of 25 units, which is insightful when assessing risk and stability across datasets of different scales.