Question

The sum of the squares of deviation for 10 observations taken from their mean 30 is 90 . The coefficient of variation is (a) \(20 \%\) (b) \(10 \%\) (c) \(11 \%\) (d) \(12 \%\)

Short answer

The coefficient of variation in this problem is found by first calculating the standard deviation, which is \(\sigma = \sqrt{\frac{90}{10}} = 3\). Then, the coefficient of variation is calculated as \(CV = \frac{3}{30} \times 100 = 10\%\). Therefore, the correct answer is (b) \(10\%\).

Step-by-step solution

Utilize the formula for standard deviation

The sum of the squares of deviation is used to find the standard deviation of a set of observations, and is found using the following formula: \[ \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}} \] where \(sigma\) (\(\sigma\)) is the standard deviation, \(x\) represents individual data points, \(\bar{x}\) is the mean of the observations, and \(N\) is the number of observations.

Calculate standard deviation from given information

In this exercise, we are given the sum of the squares of deviation (\(90\)), the mean (\(30\)), and the number of observations (\(10\)). We can now use the formula from Step 1 to find the standard deviation (\(\sigma\)): \[\sigma = \sqrt{\frac{90}{10}} \] Upon simplifying this expression, we get: \[\sigma = \sqrt{9} \] Therefore, the standard deviation is \(\sigma = 3\).

Calculate the coefficient of variation

Now that we have found the standard deviation, we can calculate the coefficient of variation. Recall that the coefficient of variation is the ratio of the standard deviation to the mean, multiplied by 100. Using the mean of 30 and the standard deviation of 3, we get: \[CV = \frac{3}{30} \times 100\] Further simplifying, we see that the coefficient of variation is \(10\%\). So the correct answer is (b) \(10\%\).

Key concepts

Standard Deviation

The Standard Deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. When we have a collection of data points, the standard deviation gives us an idea of how spread out these data points are around their mean value.

• If the data points are closer to the mean, the standard deviation will be smaller, indicating less variability.
• Conversely, if the data points are more spread out from the mean, the standard deviation will be larger, showing more variability in the data.

To calculate the standard deviation, we use the formula:\[ \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}} \] where:

• \(\sigma\) is the standard deviation.
• \(x\) represents each data point.
• \(\bar{x}\) is the mean of the observations.
• \(N\) is the total number of observations.

This formula involves calculating the sum of the squared deviations of each data point from the mean, dividing by the number of observations, and taking the square root. This step-by-step process helps to calculate how much individual data points differ from the mean, ensuring a comprehensive understanding of variability within a dataset.

Sum of Squares of Deviation

The Sum of Squares of Deviation plays a crucial role in the calculation of the standard deviation. It helps in understanding the total variability within a data set. Essentially, it is the sum of each data point's squared deviation from the mean.To break it down further:

• The deviation of each data point from the mean is computed by subtracting the mean from each observation.
• This deviation is then squared to ensure that differences do not cancel each other out. If deviations were simply added up without squaring, positive and negative deviations would cancel, giving a misleading indication of variability.

For a given set of data, the formula for the sum of squares of deviation is: \[ \sum (x - \bar{x})^2 \]In our exercise, this sum was provided as 90, for 10 observations with a mean of 30. Calculating the sum of squares of deviation is a foundational step that allows statisticians to build towards the standard deviation, offering a comprehensive picture of data spread by first looking at each point's impact on overall variability.

Mean of Observations

The Mean of Observations, often simply referred to as the mean, is a statistical measure that represents the average value of a data set. It is one of the most common and simple measures of central tendency used in statistics, providing a summary of the data set in a single value.To calculate the mean:

• You sum all the data points together.
• Then divide this total by the number of observations, \(N\).

Mathematically, it is expressed as: \[ \bar{x} = \frac{\sum x}{N} \]Where:

• \(\bar{x}\) is the mean.
• The sum \(\sum x\) denotes the addition of all data points \(x\).
• \(N\) is the number of observations.

In our particular scenario, the mean was given as 30. Understanding the mean is fundamental, as it is used to calculate other statistics like the sum of squares of deviation and the standard deviation. The mean serves as the balancing point of the data set, providing insights into where the data center lies.