Question 1

What is the difference between mean, median, and mode?

Accepted Answer

These are three distinct measures of central tendency — different answers to the question 'what is the typical value?' The mean (arithmetic average) sums all values and divides by count: it is sensitive to outliers. The median is the middle value when data is sorted — it is resistant to outliers. The mode is the most frequently occurring value — useful for categorical data. For the dataset [1, 2, 2, 3, 100]: mean = 21.6 (distorted by 100), median = 2 (robust), mode = 2 (most common). Choosing the right measure depends on data distribution and the question being asked.

Question 2

When should I use median instead of mean?

Accepted Answer

Use median when data is skewed or contains outliers that would distort the mean. Classic examples: income and wealth data (the billionaire in a room of average earners drives the mean upward while the median stays representative), home prices in mixed neighborhoods, test scores when one student dramatically outperforms others, and any biological measurement with natural outliers. The U.S. Census Bureau reports both median and mean household income specifically because they diverge — median household income was approximately $74,580 in 2022, while mean income was significantly higher due to very high earners pulling the average up.

Question 3

What does range tell you that mean and median don't?

Accepted Answer

Range (maximum minus minimum) measures variability — how spread out the data is — while mean and median only describe the center. Two datasets can have identical means and very different ranges: [10, 10, 10] and [0, 10, 20] both have mean 10 but ranges of 0 and 20 respectively. Range is the simplest measure of dispersion but also the most sensitive to outliers, since a single extreme value changes it completely. For more robust variability measurement, statisticians use interquartile range (IQR) — the range of the middle 50% — or standard deviation.

Question 4

How is standard deviation related to the average?

Accepted Answer

Standard deviation measures how far values typically deviate from the mean. It is calculated as the square root of variance — the average of squared differences from the mean. A small standard deviation means values cluster tightly around the mean; a large one means they are spread widely. In a normal distribution (bell curve), approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This '68-95-99.7 rule' is fundamental to quality control, scientific measurement, and financial risk assessment.

Question 5

What is a weighted average and when is it used?

Accepted Answer

A weighted average assigns different importance (weights) to different values before averaging. The formula is: Weighted Mean = Σ(value × weight) / Σ(weights). Common applications: GPA calculation (some courses worth 3 credits, others 4 — each grade is weighted by credit hours), investment portfolio returns (each asset's return weighted by its allocation percentage), standardized exam scores (different sections weighted differently), and quality control (defect rates weighted by production volume). A simple mean treats all values equally; a weighted mean reflects the relative importance of each data point.

Question 6

Can there be more than one mode in a dataset?

Accepted Answer

Yes. A dataset with two modes is 'bimodal'; one with more is 'multimodal.' The dataset [1, 2, 2, 3, 3, 4] has two modes: 2 and 3. Multimodal distributions often indicate the presence of multiple subpopulations in the data — for example, height data for a mixed group of adult men and women forms a bimodal distribution with peaks around male and female average heights. If every value appears exactly once, the dataset has no mode. If all values appear equally often, every value is technically a mode (which is often reported as 'no mode' in practice).

Question 7

How does sample size affect the reliability of an average?

Accepted Answer

Larger samples produce more reliable averages due to the Central Limit Theorem: as sample size increases, the sample mean approaches the true population mean, and the variability of the sample mean decreases proportionally to 1/√n. Doubling the sample size reduces the standard error of the mean by a factor of √2 ≈ 1.41. This is why polls of 1,000 people have a margin of error around ±3%, while polls of 100 people have ±10%. In practice, a sample of 30 or more observations is generally sufficient for the Central Limit Theorem to make the mean approximately normally distributed, regardless of the underlying data distribution.

Question 8

What is the difference between population mean and sample mean?

Accepted Answer

The population mean (μ) is the true average of all members of a group. The sample mean (x̄) is the average of a subset drawn from that population. The sample mean is an estimate of the population mean. Standard deviation has two versions for the same reason: population standard deviation divides by n (used when you have all data), while sample standard deviation divides by n−1 (Bessel's correction, used when working with a sample to produce an unbiased estimate of population variance). In practice, when you calculate the average of a dataset that represents a sample rather than a complete population, the n−1 denominator gives a more accurate estimate of true variability.

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