Average & Spread Calculator

Mean

The arithmetic mean (average) is the sum of values divided by the count. It is highly sensitive to outliers because every value contributes linearly.

\barx = (1)/(n) Σ_i=1^n x_i

Median

The median is the middle value after sorting: 50% of the data is below it. It is robust: moving any outlier below median or above the median does not change it. For an odd count like 5, it is the third value once sorted.

Variance

Variance is the average squared deviation from the mean. There are two common conventions:

\sigma^2 = (1)/(n) Σ (x_i - \barx)^2 \quad\text(population) \qquad s^2 = (1)/(n-1) Σ (x_i - \barx)^2 \quad\text(sample)

Why n − 1 (Bessel's correction)

When the mean itself is computed from the same data, the residuals sum to 0 — that is one constraint the deviations cannot satisfy. Dividing by n−1 instead of n "frees" the variance to be an unbiased estimator of the population variance. Bessel's correction:

s^2 = (1)/(n-1) Σ (x_i - \barx)^2

Standard deviation and range

Standard deviation is the square root of variance, returning to the original units of the data. Range is just max − min — a coarse but informative spread measure.

\sigma = √(\sigma^2), \quad s = √(s^2), \quad R = \max - \min

Choosing Between Mean, Median and Mode

The three "averages" answer different questions, and picking the wrong one is how statistics gets misused. The mean is the balance point of the data and uses every value, which makes it efficient but sensitive to outliers. The median is the middle value once sorted, unaffected by how extreme the extremes are. The mode is the most frequent value, and it is the only one that works for categorical data.

Income is the standard illustration. A handful of very high earners pulls the mean well above what a typical person earns, so median household income is the figure normally reported. The rule of thumb: use the median when the distribution is skewed or contains outliers, the mean when it is roughly symmetric, and the mode when you need the most common category rather than a central value. If mean and median differ noticeably, that gap is itself telling you the data is skewed — mean above median indicates a long right tail.

\barx = (1)/(n)Σ_i=1^n x_i

Population versus Sample

Variance comes in two versions, and the difference matters. Dividing the sum of squared deviations by n gives the population variance, correct when your data *is* the entire group. Dividing by n-1 gives the sample variance, correct when the data is a sample used to estimate a wider population.

The n-1 — Bessel's correction — exists because the deviations are measured from the sample mean rather than the true population mean, which makes them systematically a little too small. Dividing by the smaller number compensates and makes the estimator unbiased. For large n the distinction is negligible, but for small samples it is substantial: with n = 5 it changes the variance by 25%. Statistical software defaults vary, so always check which one you are getting before comparing results across tools.

\sigma^2 = (Σ (x_i - \mu)^2)/(n), \qquad s^2 = \fracΣ (x_i - \barx)^2n-1

Weighted and Specialised Means

When observations carry different importance, the weighted mean multiplies each value by its weight before averaging. A course grade combining a 30% midterm and 70% final is exactly this calculation, as is a portfolio return weighted by holding size.

Two other means solve problems the arithmetic mean gets wrong. The geometric mean — the n-th root of the product — is the correct average for growth rates and ratios: an investment that gains 50% then loses 50% has an arithmetic mean return of 0% but a geometric mean of −13.4%, and the geometric figure is the one that matches the actual ending balance. The harmonic mean is the right choice for averaging rates over a fixed distance: driving 60 km/h out and 40 km/h back gives an average speed of 48 km/h, not 50. For any set of positive numbers, harmonic ≤ geometric ≤ arithmetic, with equality only when all values are identical.

\barx_w = (Σ w_i x_i)/(Σ w_i), \quad G = \sqrt[n]\prod x_i, \quad H = (n)/(Σ 1/x_i)

Worked Example

Inputs

• x1: 2
• x2: 4
• x3: 6
• x4: 8
• x5: 10

Result: mean: 6 median: 6 sampleVariance: 10 sampleStd: 3.1623 range: 8

Explanation

Mean = (2+4+6+8+10)/5 = 6. Median = 6. Sample variance s² = ((2−6)² + (4−6)² + (6−6)² + (8−6)² + (10−6)²)/4 = 40/4 = 10. Sample std s = √10 ≈ 3.162. Range = 10 − 2 = 8.

Second Scenario

Inputs

• x1: 3.5
• x2: 4
• x3: 6
• x4: 8
• x5: 10

Result: mean: 6 median: 6 sampleVariance: 10 sampleStd: 3.1623 range: 8

Explanation

This scenario uses different inputs (x1 = 3.5, x2 = 4, x3 = 6, x4 = 8, x5 = 10) to show how changing one variable affects the average & spread result. Run the calculator above with these values to get the exact updated output with step-by-step work.