Calculate The Sample Standard Deviation of The Following Four Prices

What is Standard Deviation?

Standard deviation (SD) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

There are two main types of standard deviation:

• Population standard deviation - Calculated using the entire population of data.
• Sample standard deviation - Calculated using a sample of data taken from a larger population.

For this guide, we'll focus on calculating the sample standard deviation, which is more commonly used when working with samples rather than entire populations.

How to Calculate Sample Standard Deviation

The formula for calculating the sample standard deviation (s) is:

Where:

• s = sample standard deviation
• Σ = sum of
• xi = each individual value in the data set
• x̄ = sample mean
• n = number of observations in the sample

The steps to calculate the sample standard deviation are:

1. Calculate the sample mean (x̄) by summing all values and dividing by the number of observations.
2. For each data point, subtract the sample mean and square the result (the squared difference).
3. Sum all of the squared differences.
4. Divide the sum of squared differences by n - 1 (where n is the number of observations).
5. Take the square root of the result to get the sample standard deviation.

Example Calculation

Let's calculate the sample standard deviation for the following four prices: $10, $12, $14, and $16.

1. Calculate the sample mean:
   x̄ = (10 + 12 + 14 + 16) / 4 = 62 / 4 = $15.50
2. Calculate each squared difference from the mean:
   • (10 - 15.5)² = (-5.5)² = 30.25
   • (12 - 15.5)² = (-3.5)² = 12.25
   • (14 - 15.5)² = (-1.5)² = 2.25
   • (16 - 15.5)² = (0.5)² = 0.25
3. Sum the squared differences:
   Σ(xi - x̄)² = 30.25 + 12.25 + 2.25 + 0.25 = 45
4. Divide by n - 1 (degrees of freedom):
   45 / (4 - 1) = 45 / 3 = 15
5. Take the square root to get the standard deviation:
   s = √15 ≈ $3.87

The sample standard deviation of these four prices is approximately $3.87.

Interpreting the Results

The sample standard deviation provides several important insights:

• Measure of dispersion: A higher standard deviation indicates that the data points are more spread out from the mean.
• Relative to the mean: The standard deviation is in the same units as the original data, making it easy to interpret.
• Comparison tool: You can compare standard deviations of different data sets to understand which has more variability.

For example, if you have two sets of prices with the same mean but different standard deviations, the set with the higher standard deviation would have prices that vary more widely from the average price.