Calculate Sample Mean and Standard Deviation
Compute the sample mean and sample standard deviation from a dataset of exam scores. This foundational skill is used in nearly every area of statistics.
Problem Scenario
A statistics instructor collected the following exam scores (out of 100) from a random sample of 8 students in her introductory course: 72, 85, 91, 68, 77, 83, 95, 89. She wants to summarize the class performance by computing the sample mean and sample standard deviation.
Given Data
Requirements
- Calculate the sample mean of the dataset
- Calculate the sample standard deviation using the correct formula (dividing by n - 1)
- Interpret the results in context
Solution
Step 1:
Calculate the sample mean: x-bar = (sum of all values) / n = (72 + 85 + 91 + 68 + 77 + 83 + 95 + 89) / 8 = 660 / 8 = 82.5.
Step 2:
Find each deviation from the mean: (72 - 82.5) = -10.5, (85 - 82.5) = 2.5, (91 - 82.5) = 8.5, (68 - 82.5) = -14.5, (77 - 82.5) = -5.5, (83 - 82.5) = 0.5, (95 - 82.5) = 12.5, (89 - 82.5) = 6.5.
Step 3:
Square each deviation: 110.25, 6.25, 72.25, 210.25, 30.25, 0.25, 156.25, 42.25. Sum of squared deviations = 628.00.
Step 4:
Calculate the sample variance: s^2 = sum of squared deviations / (n - 1) = 628.00 / 7 = 89.71 (rounded to two decimal places).
Step 5:
Calculate the sample standard deviation: s = sqrt(89.71) = 9.47 (rounded to two decimal places). This tells us the typical exam score deviates about 9.47 points from the mean of 82.5.
Final Answer
The sample mean is 82.5 and the sample standard deviation is approximately 9.47. On average, students scored 82.5 out of 100, with scores typically varying by about 9.47 points from the mean.
Key Takeaways
- ✓The sample standard deviation uses n - 1 in the denominator (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
- ✓The mean is sensitive to outliers; always examine your data for extreme values before relying solely on the mean.
- ✓Standard deviation is in the same units as the original data, making it directly interpretable.
Common Errors to Avoid
- ✗Dividing by n instead of n - 1 when computing sample standard deviation. The n - 1 correction accounts for the lost degree of freedom when estimating the population mean from sample data.
- ✗Forgetting to take the square root of the variance. The variance is in squared units; the standard deviation brings it back to the original scale.
- ✗Arithmetic errors when computing deviations from the mean. Double-check that the sum of all deviations equals zero (within rounding).
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Common questions about this problem type
Use n when computing the population standard deviation (when you have data for the entire population). Use n - 1 when computing the sample standard deviation (when your data is a sample drawn from a larger population). In most homework and real-world problems you are working with samples, so n - 1 is the correct choice.
The standard deviation measures the average amount of spread or dispersion in your dataset. A small standard deviation means data points cluster tightly around the mean, while a large standard deviation means they are more spread out. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean.