Question

Calculate the SD of each of the following fictitious samples: (a) 8,6,9,4,8 (b) 4,7,5,4 (c) 9,2,6,7,6

Short answer

SD (a) = 2, SD (b) = \( \sqrt{2} \), SD (c) = \( \sqrt{6.5} \).

Step-by-step solution

Calculate the Mean for Sample (a)

Firstly, sum all the numbers and then divide by the quantity of the numbers. Mean (a) = (8 + 6 + 9 + 4 + 8) / 5 = 35 / 5 = 7.

Find the Deviations from the Mean for Sample (a)

Subtract the mean from each number to find the deviations: 8-7, 6-7, 9-7, 4-7, 8-7, which gives 1, -1, 2, -3, 1.

Square the Deviations for Sample (a)

Square each deviation: 1^2, (-1)^2, 2^2, (-3)^2, 1^2 which gives 1, 1, 4, 9, 1.

Calculate the Variance for Sample (a)

Sum the squared deviations and divide by the number of values minus one (n-1). Variance (a) = (1 + 1 + 4 + 9 + 1) / (5 - 1) = 16 / 4 = 4.

Calculate the Standard Deviation for Sample (a)

Take the square root of the variance. SD (a) = \( \sqrt{4} \) = 2.

Calculate the Mean for Sample (b)

Find the mean of the numbers. Mean (b) = (4 + 7 + 5 + 4) / 4 = 20 / 4 = 5.

Find the Deviations from the Mean for Sample (b)

Subtract the mean from each number: 4-5, 7-5, 5-5, 4-5 which gives -1, 2, 0, -1.

Square the Deviations for Sample (b)

Square each deviation: (-1)^2, 2^2, 0^2, (-1)^2 which gives 1, 4, 0, 1.

Calculate the Variance for Sample (b)

Sum the squared deviations and divide by the number of values minus one (n-1). Variance (b) = (1 + 4 + 0 + 1) / (4 - 1) = 6 / 3 = 2.

Calculate the Standard Deviation for Sample (b)

Take the square root of the variance. SD (b) = \( \sqrt{2} \).

Calculate the Mean for Sample (c)

Find the mean of the numbers. Mean (c) = (9 + 2 + 6 + 7 + 6) / 5 = 30 / 5 = 6.

Find the Deviations from the Mean for Sample (c)

Subtract the mean from each number: 9-6, 2-6, 6-6, 7-6, 6-6 which gives 3, -4, 0, 1, 0.

Square the Deviations for Sample (c)

Square each deviation: 3^2, (-4)^2, 0^2, 1^2, 0^2 which gives 9, 16, 0, 1, 0.

Calculate the Variance for Sample (c)

Sum the squared deviations and divide by the number of values minus one (n-1). Variance (c) = (9 + 16 + 0 + 1 + 0) / (5 - 1) = 26 / 4 = 6.5.

Calculate the Standard Deviation for Sample (c)

Take the square root of the variance. SD (c) = \( \sqrt{6.5} \).

Key concepts

Statistical Mean

The statistical mean, often simply called the mean or average, is a central concept in statistics and represents the typical value in a set of numbers. It's calculated by adding together all the values in a dataset and then dividing by the number of values.

For instance, in the solution provided for Sample (a) with the numbers 8, 6, 9, 4, 8, we calculate the mean by summing these numbers to get 35 and dividing by the quantity of numbers, which is 5. This gives us a mean of 7. The mean is akin to the center of gravity for a dataset, and each data point contributes to the calculation equally.

Finding the mean is the preliminary step in calculating both variance and standard deviation, as it serves as the reference point from which we measure each data point's deviation.

Squared Deviations

Squared deviations reflect how far each value in a dataset is from the mean. To determine squared deviations, we first find the difference between each value and the mean, which gives us the deviation. Then, we square each of these deviations.

Squaring is necessary because it achieves two things: it removes any negative signs, which would otherwise cancel out variance when summed, and it gives more weight to larger deviations, emphasizing outliers. For example, in Sample (a), after finding the mean (7), we calculate the deviations (e.g., 8 - 7 = 1) and then square them (e.g., 1^2 = 1), resulting in 1, 1, 4, 9, 1.

This process highlights the dispersion in the dataset and sets the stage for finding the overall measure of spread, which is the variance.

Variance

Variance is a statistical measurement that describes the spread of the numbers in a dataset. It is calculated by taking the average of the squared deviations. When calculating the sample variance, as opposed to the population variance, we divide the sum of squared deviations by one fewer than the number of values in the dataset, called the sample size minus one (n - 1). This adjustment, known as Bessel's correction, corrects the bias in the estimation of the population variance from a sample.

Continuing with Sample (a), the squared deviations were 1, 1, 4, 9, 1. We sum these up to get 16 and then divide by 4 (which is one less than the number of data points, 5), obtaining a variance of 4. A higher variance indicates a greater spread of the numbers around the mean. Conversely, a small variance indicates that the numbers are closer to the mean.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of the average distance between each data point and the mean. It's denoted as SD and is expressed in the same units as the data, making it more interpretable than variance.

For Sample (a), the variance was 4. Taking the square root of 4 gives us a standard deviation of 2. This means that, on average, the data points in Sample (a) deviate by 2 units from the mean of 7. The standard deviation is an essential tool for statisticians and researchers because it provides a way to assess variability in data and to compare the spread between different datasets or within the same dataset over time.