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Mean Median Mode and Range Formulas

Review arithmetic mean, odd and even median, mode frequency, range, and midrange formulas with variables, calculation order, and a worked example.

How do the Mean Median Mode Calculator formulas work?

This formula page defines the variables used by the Mean Median Mode Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

Mean, median, and mode are separate measures of center. The mean uses the magnitude of every observation, the median uses ordered position, and the mode uses frequency. Range and midrange summarize the endpoints.

Variable Definitions

  • xᵢ is observation i.
  • n is the number of observations.
  • Σxᵢ is the sum of all observations.
  • x₍ᵢ₎ is the value in position i after sorting.
  • f(x) is the frequency of value x.
  • x min and x max are the minimum and maximum values.

Arithmetic Mean Formula

xˉ=∑i=1nxin\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}

Add every observation, then divide by the number of observations. Because every value contributes to the sum, an unusually large or small observation can noticeably change the mean.

Median Formulas

First sort the values as x₍₁₎ ≤ x₍₂₎ ≤ … ≤ x₍ₙ₎.

For an odd number of observations:

Median⁡=x((n+1)/2)\operatorname{Median}=x_{((n+1)/2)}

For an even number of observations:

Median⁡=x(n/2)+x(n/2+1)2\operatorname{Median}=\frac{x_{(n/2)}+x_{(n/2+1)}}{2}

Mode Frequency Definition

Mode⁡={x:f(x)=max⁡yf(y)}\operatorname{Mode}=\{x:f(x)=\max_y f(y)\}

The set includes every value tied for the highest frequency. This calculator reports no mode when all distinct values occur equally often.

Range and Midrange Formulas

Range⁡=xmax⁡−xmin⁡\operatorname{Range}=x_{\max}-x_{\min}
Midrange⁡=xmin⁡+xmax⁡2\operatorname{Midrange}=\frac{x_{\min}+x_{\max}}{2}

Range describes the distance between the endpoints. Midrange locates their midpoint but ignores every observation between them.

Calculation Order

  1. Parse and sort the observations.
  2. Add and count the values for the mean.
  3. Select or average the middle position for the median.
  4. Count exact repeated values for the mode.
  5. Use the minimum and maximum for range and midrange.
  6. Round displayed decimal results only at the end.

Worked Calculation

For 1, 1, 2, 2, 3, the sum is 9 and n is 5, so the mean is 9 ÷ 5 = 1.80. The third ordered observation is 2, so the median is 2. Values 1 and 2 each occur twice, so the data are bimodal. The range is 3 - 1 = 2, and the midrange is (1 + 3) ÷ 2 = 2.

Formula Assumptions

  • All observations are numeric and use one consistent unit.
  • Each observation has equal weight.
  • Frequency uses exact values rather than rounded groups or intervals.
  • Missing observations are excluded rather than inferred.
  • Displayed values are rounded to two or three decimal places.

Formula FAQ

Why must the data be sorted for the median?

The median is defined by ordered position. Without sorting, the middle entry in the original list may not be the middle value. Use realistic inputs and review the Mean Median Mode Calculator formula assumptions before relying on this result for decisions.

Can a data set have several modes?

Yes. Every value tied for the highest frequency is a mode. For a useful Mean Median Mode Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Does range use every observation?

No. Range uses only the minimum and maximum, which is why it can change sharply when an endpoint changes. For a useful Mean Median Mode Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

Next Steps

Apply these formulas in the Mean Median Mode Calculator, follow the verified examples, or measure variation using the Standard Deviation Calculator.

Use the Mean Median Mode Calculator