Question

You are given \(n=8\) measurements: 3,1,5,6 4,4,3,5 a. Calculate the range. b. Calculate the sample mean. c. Calculate the sample variance and standard deviation. d. Compare the range and the standard deviation. The range is approximately how many standard deviations?

Short answer

Answer: Approximately 3.22 standard deviations are equivalent to the range of the given measurements.

Step-by-step solution

Find the range

First, we need to identify the highest and lowest measurements. In the given measurements, the highest value is 6 and the lowest value is 1. The range is the difference between the highest and lowest values: Range = 6 - 1 = 5.

Calculate the sample mean

Next, we need to find the sample mean. To do this, we will add all the measurements and divide by the total number of measurements (8): Mean = (3+1+5+6+4+4+3+5) / 8 = 31 / 8 = 3.875.

Calculate the sample variance

To find the sample variance, we will first find the squared differences from the mean for each measurement: (3-3.875)^2, (1-3.875)^2, (5-3.875)^2, (6-3.875)^2, (4-3.875)^2, (4-3.875)^2, (3-3.875)^2, and (5-3.875)^2. Then, we will find the average of these squared differences: Variance = [(0.765625 + 8.265625 + 1.265625 + 4.515625 + 0.015625 + 0.015625 + 0.765625 + 1.265625) / (8-1)] = 16.875 / 7 = 2.4107142857.

Calculate the sample standard deviation

To find the sample standard deviation, we will take the square root of the sample variance: Standard Deviation = √2.4107142857 = 1.5514 (rounded to 4 decimal places).

Compare the range and standard deviation

Now that we have both the range (5) and the standard deviation (1.5514), we can compare the two values to see how many standard deviations are approximately equivalent to the range: 5 / 1.5514 ≈ 3.22. The range is approximately 3.22 standard deviations.

Key concepts

Range

The range of a dataset is the simplest measure of variability, showing the spread between the highest and lowest observation. Identifying the range is straightforward. To calculate the range for a set of measurements, follow these steps:

• Find the largest number in your dataset.
• Identify the smallest number in your dataset.
• Subtract the smallest number from the largest to yield the range.

For the given set of numbers: 3, 1, 5, 6, 4, 4, 3, and 5, the largest number is 6, and the smallest is 1. Thus, the range is calculated as 6 - 1 = 5.
This calculation helps quickly understand the overall spread of the data, although it doesn’t provide insights into the distribution of values within this range.

Sample Mean

The sample mean is a measure of the central tendency of a dataset, essentially providing an average of the given measurements. It helps to determine the "typical" value. To compute the sample mean:

• Add together all the measurements.
• Divide the sum by the total number of observations in the sample (n).

For the given values 3, 1, 5, 6, 4, 4, 3, and 5, the sum is 31. Since there are 8 measurements, we calculate the sample mean by dividing 31 by 8, resulting in a mean of 3.875.
In summary, the sample mean helps to understand the general magnitude of the dataset, but it’s important to remember that it can be affected by extreme values or outliers.

Sample Variance

Sample variance measures how much the individual data points in a sample differ from the sample mean. It's a crucial component in understanding the data's volatility. To calculate the sample variance:

• Subtract the sample mean from each measurement to get the deviation from the mean.
• Square each of these deviations to eliminate negative differences.
• Calculate the average of these squared deviations, dividing by one less than the sample size (n-1).

From the previous example, once each deviation is squared, they sum to 16.875. Dividing this sum by 7 (one less than the number of observations) gives a sample variance of approximately 2.41.
The sample variance provides insight into the data’s dispersion and is vital for deriving the sample standard deviation.

Sample Standard Deviation

The sample standard deviation is a widely used measure of data dispersion, being the square root of the sample variance. It presents the spread of a dataset in the same unit as the original data, making it easier to interpret. To find the sample standard deviation:

• First, calculate the sample variance.
• Take the square root of the sample variance.

In our dataset, with a sample variance of approximately 2.41, the sample standard deviation becomes approximately 1.55 when we take the square root.
This measure indicates the average amount by which each data point differs from the mean. In practical terms, it allows for an intuitive grasp of how "spread out" the values are. When comparing with other measures like the range, standard deviation provides a sense of consistency in variability across the dataset.