Question

Find the mean, standard deviation and variance of the following data set using both methods for standard deviation and variance: The definition of standard deviation. The computational formula 48 36 38 40 43 45 46 47 45.

Answer 1

To find the mean of the given data set, sum all the numbers and divide by their count. The standard deviation can be found using either the definition or the computational formula, both involving deviations from the mean. A calculator or computer software is commonly used to simplify these calculations and provide insights into the spread of the data.

Step-by-step explanation:

To find the mean, add up all the numbers and divide by how many there are. Our data set is 48, 36, 38, 40, 43, 45, 46, 47, 45. The mean (also called the average) = (48 + 36 + 38 + 40 + 43 + 45 + 46 + 47 + 45) / 9 = 388 / 9 ≈ 43.1.

For the standard deviation (SD), there are two formulas: the definition formula and the computational formula. Using the definition of standard deviation, calculate each deviation from the mean (value - mean), square it, sum these squares, divide by the number of data points, and then take the square root:

1. Calculate deviations: (48 - 43.1)^2, (36 - 43.1)^2, (38 - 43.1)^2, etc.
2. Sum the squares of deviations.
3. Divide by the number of data points (n=9 for a population, n-1=8 for a sample) to get the variance.
4. Take the square root of variance to obtain standard deviation.

The computational formula for the variance simplifies the process by subtracting the square of the mean from the mean of the squares of values. The standard deviation is simply the square root of this variance.

For both formulas, you can use a calculator or computer software for easier computation, and it's essential to understand that the standard deviation measures how spread out the numbers are around the mean.