Source: ENGR 216 Practice Challenges, Texas A&M
Tags: mean, median, mode, range, standard deviation, variance, descriptive statistics, central tendency, dispersion, ENGR 216, experimental physics
Descriptive statistics condense a dataset into a handful of numbers that tell you where the data centres and how spread out it is. For ENGR 216, you need to compute the mean, median, mode, range, standard deviation, and variance by hand (non-programmable calculator only), and you need to do it quickly under timed conditions.
Mean (arithmetic mean)
The sum of all data values divided by the number of values. The most common measure of central tendency.
Median
The middle value when data are arranged in ascending order. For an even number of values, take the average of the two middle values.
Mode
The value that appears most frequently. A dataset can have one mode, multiple modes, or no mode at all.
Range
The difference between the largest and smallest values in the dataset. A rough measure of spread.
Standard deviation (s)
A measure of how far individual data points typically fall from the mean. The square root of the variance.
Variance (s²)
The average of the squared deviations from the mean. For a sample, divide by (n − 1), not n.
Population vs. sample
If the dataset represents an entire population, divide by N. If it is a sample drawn from a larger population, divide by (n − 1). In ENGR 216 practice problems, assume sample statistics unless told otherwise.
Sum every value in the dataset.
Divide by n (the count of values).
Formula: x̄ = (Σxᵢ) / n
Sort all values from smallest to largest.
If n is odd, the median is the value at position (n + 1) / 2.
If n is even, the median is the average of the values at positions n/2 and (n/2) + 1.
Tally how often each distinct value appears.
The value with the highest frequency is the mode.
If multiple values share the highest frequency, report all of them (multimodal).
Identify the maximum and minimum values.
Range = max − min
Find the mean (x̄).
Subtract the mean from each data value to get deviations: (xᵢ − x̄).
Square each deviation: (xᵢ − x̄)².
Sum all squared deviations: Σ(xᵢ − x̄)².
Divide by (n − 1) for sample variance.
Take the square root for sample standard deviation.
Mean: x̄ = (Σxᵢ) / n
Sample variance: s² = Σ(xᵢ − x̄)² / (n − 1)
Sample standard deviation: s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
Range: R = x_max − x_min
Dataset (25 values):
8.8, 9.5, 9.8, 9.4, 10.0, 9.4, 10.1, 9.2, 11.3, 9.4, 10.0, 10.4, 7.9, 10.4, 9.8, 9.8, 9.5, 8.9, 8.8, 10.6, 10.1, 9.5, 9.6, 10.2, 8.9
Step 1: Sort the data
7.9, 8.8, 8.8, 8.9, 8.9, 9.2, 9.4, 9.4, 9.4, 9.5, 9.5, 9.5, 9.6, 9.8, 9.8, 9.8, 10.0, 10.0, 10.1, 10.1, 10.2, 10.4, 10.4, 10.6, 11.3
Step 2: Mean
Sum = 8.8 + 9.5 + 9.8 + 9.4 + 10.0 + 9.4 + 10.1 + 9.2 + 11.3 + 9.4 + 10.0 + 10.4 + 7.9 + 10.4 + 9.8 + 9.8 + 9.5 + 8.9 + 8.8 + 10.6 + 10.1 + 9.5 + 9.6 + 10.2 + 8.9 = 241.3
x̄ = 241.3 / 25 = 9.65
Step 3: Median
25 values, so the median is the 13th value in the sorted list.
Median = 9.6 (the 13th value)
Step 4: Mode
9.4 appears 3 times, 9.5 appears 3 times, 9.8 appears 3 times.
Mode = 9.4, 9.5, and 9.8 (trimodal)
Step 5: Range
Range = 11.3 − 7.9 = 3.40
Step 6: Variance and standard deviation
Compute each (xᵢ − x̄)², sum them, divide by 24.
Σ(xᵢ − x̄)² = 10.494
s² = 10.494 / 24 = 0.44
s = √0.44 = 0.66
⚠️ Use (n − 1) in the denominator for sample variance and standard deviation, not n. This is the most common calculator mistake under time pressure.
⚠️ When finding the median, you must sort first. Skipping the sort is an easy error when rushed.
⚠️ A dataset can have more than one mode. If the problem asks for "the mode," list all values that share the highest frequency.
⚠️ Round only at the final answer, not at intermediate steps. Premature rounding cascades error.
⚠️ The exam says "round to two decimal places" and "do not include units." Follow those instructions exactly.
Q: What is the difference between population standard deviation and sample standard deviation?
A: Population standard deviation divides by N (the full population count). Sample standard deviation divides by (n − 1) to correct for the bias introduced by estimating the population mean from a sample.
Q: A dataset has values 3, 5, 5, 7, 9. What are the mean, median, and mode?
A: Mean = 29/5 = 5.80. Median = 5 (the 3rd value). Mode = 5 (appears twice, more than any other value).
Q: Why does variance use squared deviations rather than simple deviations?
A: Simple deviations from the mean always sum to zero (positive and negative cancel). Squaring ensures all terms are positive, producing a meaningful measure of spread.
Q: If every value in a dataset is the same, what are the variance and standard deviation?
A: Both are zero. There is no spread around the mean.
descriptive statistics, central tendency, dispersion, spread, arithmetic mean, sample mean, population mean, median, mode, unimodal, bimodal, multimodal, range, variance, standard deviation, sum of squares, degrees of freedom, n minus 1, ENGR 216, experimental physics lab, Texas A&M