Statistics Calculator (Mean, SD)

Statistics Measures Guide

What should I know about Mean, Median, and Mode?

Mean: sum of all values ÷ count. Sensitive to outliers — a single extreme value can significantly shift the mean. Median: the middle value when sorted. For even counts, the average of the two middle values. Much more resistant to outliers — the median income is more representative of typical earnings than mean income because a few extremely high earners pull the mean up significantly. Mode: the most frequently occurring value. Can be multiple (bimodal distribution) or none (all values unique). Median is generally the more representative average for skewed data, such as income or house prices, where a small number of high values would otherwise distort the mean.

What's the key thing to understand about Standard Deviation?

Standard deviation measures how spread out values are from the mean. A small SD means values cluster tightly around the mean. A large SD means high variability. Population SD divides by n; sample SD divides by n−1 (Bessel's correction — accounts for the fact that a sample underestimates population variance). Approximately 68% of values fall within 1 SD of the mean in a normal distribution, 95% within 2 SD, 99.7% within 3 SD (the empirical rule / 68-95-99.7 rule).

When to Use Which Measure?

Mean: best for symmetrical distributions without outliers. Median: best for skewed distributions or data with outliers (income, house prices, response times). Mode: best for categorical data or finding the most common value. Use both mean and median when you have a dataset — if they differ significantly, you have a skewed distribution or outliers. For investment returns, the geometric mean (not arithmetic) is the correct measure of average compounding growth.

What do I need to know about Five-Number Summary?

The five-number summary (minimum, Q1, median, Q3, maximum) gives a comprehensive view of distribution shape. Q1 is the median of the lower half; Q3 is the median of the upper half. The interquartile range (IQR = Q3 − Q1) captures the middle 50% of data. Outliers are often defined as values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. Box plots visualise the five-number summary and are particularly useful for comparing distributions across groups.

What's the key thing to understand about Skewness?

Skewness describes the asymmetry of a dataset's distribution and shows up as a predictable relationship between the mean and the median. In a right-skewed (positively skewed) distribution, a small number of unusually large values pull the mean upward while the median stays closer to the bulk of typical values, so mean > median — household income is a classic real-world example, where a few very high earners drag the average above the typical person's earnings. In a left-skewed (negatively skewed) distribution, the opposite happens: a small number of unusually small values pull the mean downward, so mean < median — exam scores on an easy test, where most students score highly but a few very low scores drag the average down, often show this pattern. In a perfectly symmetric distribution, such as an ideal normal distribution, the mean and median are equal. Comparing your calculated mean and median is therefore a quick, practical way to detect skewness without needing a more advanced skewness formula or a chart.

What should I know about Outliers and Their Effect on Statistics?

An outlier is a data point that lies unusually far from the rest of a dataset, and different summary statistics react very differently to its presence. The mean is highly sensitive to outliers because every value, including extreme ones, is included in the sum that gets divided by the count — a single very large or very small value can shift the mean substantially. The median is much more robust (resistant to outliers), since it only depends on which value sits in the middle position, largely ignoring how extreme the values at either end actually are. A standard method for formally identifying outliers uses the interquartile range (IQR = Q3 − Q1, from the five-number summary): any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is flagged as a statistical outlier. This is exactly why the choice between mean and median matters so much in practice — financial and real estate data, which often include extreme outliers, are usually better summarised by the median, while more tightly clustered data is well-represented by the mean.

What's the key thing to understand about Population Parameters vs Sample Statistics?

Statistics draws an important distinction between a population — every member of the group you ultimately care about — and a sample, a smaller subset actually measured. Values describing the whole population are called parameters and conventionally use Greek letters, like μ (population mean) and σ (population standard deviation), while the corresponding values calculated from a sample are called statistics and use Roman letters, like x̄ (sample mean) and s (sample standard deviation). This distinction matters because sample statistics are estimates of the true population parameters, and they carry uncertainty that shrinks as the sample gets larger — which is exactly why the sample standard deviation formula divides by (n-1) rather than n, a small adjustment (called Bessel's correction) that corrects for the tendency of a sample to slightly underestimate the true population variability.

When Should You Report Range, IQR, or Standard Deviation as the Spread Measure?

Spread (how dispersed a dataset is) can be summarized several ways, and picking the wrong one for the data's shape gives a misleading impression of consistency. Range (max minus min) is the simplest but is entirely determined by just two extreme values, making it highly sensitive to outliers. The interquartile range (IQR, the spread of the middle 50% of data) ignores extreme values entirely, making it the most robust choice for skewed data or data with outliers. Standard deviation uses every data point and is the standard choice for roughly symmetric, outlier-free data, especially when further statistical tests assume it.

Worked example: a dataset of 9 house prices in thousands — 180, 190, 195, 200, 205, 210, 215, 220, 650 — has a range of 470 (650-180), dominated entirely by the one unusual 650 value. The IQR, however, captures the typical spread without that distortion: sorting gives a median of 205, a lower quartile around 195, and an upper quartile around 217.5, for an IQR of about 22.5 — a far more representative picture of how tightly clustered most of the houses actually are, with the $650k property correctly treated as an outlier rather than allowed to dominate the spread measure.

What Is a Z-Score and How Does It Relate to Standard Deviation?

A z-score expresses how many standard deviations a particular value sits from the mean, calculated as z = (x - mean) / standard deviation. It standardizes values from different datasets or scales onto a common footing, which is why z-scores are used to compare a student's performance across two tests with different score ranges, or to flag unusually extreme values in a dataset regardless of the data's original units. A z-score of 0 means the value equals the mean exactly; a z-score of +2 means the value sits two standard deviations above the mean, which — for roughly normally distributed data — places it in the top 2.3% or so of the distribution, since the 68-95-99.7 rule puts about 95% of values within two standard deviations of the mean in either direction.

Worked Example: Calculating Mean, Median and Standard Deviation

Take the dataset: 4, 8, 6, 5, 3, 9, 7. The mean is the sum (42) divided by the count (7), giving 6.0. Sorting the data (3, 4, 5, 6, 7, 8, 9), the median is the middle value, also 6.0. For standard deviation, first find each value's deviation from the mean and square it: (4-6)^2=4, (8-6)^2=4, (6-6)^2=0, (5-6)^2=1, (3-6)^2=9, (9-6)^2=9, (7-6)^2=1. Summing these gives 28. Dividing by n (7) for the population variance gives 4.0, so the population standard deviation is the square root of 4.0, which is exactly 2.0. If this were a sample rather than the full population, you would divide by (n-1)=6 instead, giving a sample variance of 4.67 and a sample standard deviation of about 2.16 — slightly larger, which reflects the extra uncertainty of estimating spread from a sample rather than knowing it from complete population data.