Question

The sum of 10 observation is 150 and the sum of their squares is 2700 . The standard deviation is (a) \(3 \sqrt{5}\) (b) \(5 \sqrt{3}\) (c) 15 (d) 5

Short answer

Given the sum of 10 observations is 150, the mean \(\bar{x} = \frac{150}{10} = 15\). The sum of the squares of the observations is 2700. To find the variance, \(s^2 = \frac{\sum {x^2}}{N} - \bar{x}^2\), where N = number of observations. Substitute values: \(s^2 = \frac{2700}{10} - 15^2 = 45\). To find the standard deviation, \(s = \sqrt{s^2}\), so \(s = \sqrt{45} = 3\sqrt{5}\). Hence, the correct answer is (a) \(3\sqrt{5}\).

Step-by-step solution

Calculate the mean of the observations

Given that the sum of 10 observations is 150: Mean, \(\bar{x} = \frac{\text{Sum of observations}}{\text{Number of observations}}\) \( \bar{x} = \frac{150}{10}\) \( \bar{x} = 15\)

Find the variance of the observations

We know the sum of the squares of the observations is 2700. We'll use the formula for variance: Variance, \(s^2 = \frac{\sum {x^2}}{N} - \bar{x}^2\) where \(\sum{x^2}\) = sum of the squares of the observations, N = number of observations, and \(\bar{x}\) = mean Substitute the given values into the equation: \(s^2 = \frac{2700}{10} - 15^2\) \(s^2 = 270 - 225\) \(s^2 = 45\)

Calculate the standard deviation

The standard deviation is the square root of the variance: Standard Deviation, \(s = \sqrt{s^2}\) \(s = \sqrt{45}\) \(s = 3\sqrt{5}\) So, the correct answer is (a) \(3\sqrt{5}\).

Key concepts

Variance

Variance is a statistical measurement that indicates the spread of a set of numbers. It's a way of looking at how much the data points in a set differ from the mean (average) value.

Think of it like this: if all your observations were exactly the same, there would be no variance at all, because every number would be equal to the mean. Variance comes into play when there's diversity or spread in the data. The greater the spread, the higher the variance.

To calculate variance, you would generally subtract the mean from each observation, square the result to make it positive (since we don't care about the direction but the magnitude of the deviation), and then average those squared differences. This is known as the 'mean of squared deviations from the mean'.

In the context of the provided exercise, after finding the mean to be 15, variance was determined using a slightly different approach, by subtracting the square of the mean from the average of the squares of the observations. This method simplifies the calculation and avoids computing the square of deviations for each individual observation.

Mean of Observations

The mean of observations, often simply called the mean, is the average value of a set of numbers. It gives you an idea of what the typical value in your dataset might be.

To find the mean, you total up all the observations and then divide by the number of observations you have. In the realm of statistics, the mean is like a center of gravity, balancing the data set on a scale; it pulls together all the numbers to find their collective center.

In our exercise, the sum of 10 observations was 150, so the mean was calculated as \( \bar{x} = \frac{150}{10} \), which simplifies to 15. This means that on average, each observation in this dataset would be 15 if all values were the same. However, we know that's not the case, hence the need for further statistical measures like variance and standard deviation to understand the data's dispersion.

Sum of Squares

The term 'sum of squares' sounds like it should be straight out of a math-centric workout routine, but in statistics, it's a critical concept for measuring variation. It involves squaring each data observation, which amplifies discrepancies, and then adding all of these squares up. It's like saying, 'Let's emphasize our differences, and then see how much they amount to collectively.'

This approach is particularly powerful because squaring eliminates any negative signs—only the magnitude of the deviation matters, not whether it's above or below the mean.

In our example, the sum of squares was given as 2700. This number doesn't tell us much by itself, but when used along with the mean in the formula for variance, it turns into an essential piece of the statistical puzzle, leading to the identification of variance as 45. When the square root of this variance is taken, we arrive at the standard deviation, which in this case was \(3\sqrt{5}\), providing a measure of spread that incorporates the magnitude of each observation's deviation from the mean.