Central tendency in descriptive statistics
A mean median mode calculator reports three standard measures of central tendency for ungrouped data. Each measure locates the “center” in a different mathematical sense: average value (mean), central ordered position (median), and most frequent value (mode).
Definitions and formulas
| Measure | Symbol | Definition | Formula (ungrouped data) |
|---|---|---|---|
| Mean | \(\bar{x}\) | Arithmetic average; “balance point” of the data on a number line. | \(\displaystyle \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i\) |
| Median | \(\tilde{x}\) | Middle value after sorting; splits the data into two halves. |
Odd \(n\): \(\tilde{x}=x_{((n+1)/2)}\) Even \(n\): \(\tilde{x}=\dfrac{x_{(n/2)}+x_{(n/2+1)}}{2}\) |
| Mode | — | Most frequent value; tied maxima produce multiple modes. | Value(s) with maximum frequency. |
Visualization of mean, median, and mode with and without an outlier
Worked example (ungrouped data)
Dataset A: \(2, 3, 3, 4, 10\). The ordered list is \(2 \le 3 \le 3 \le 4 \le 10\), with \(n=5\).
\[ \bar{x}=\frac{2+3+3+4+10}{5}=\frac{22}{5}=4.4 \]
\[ \tilde{x}=x_{((5+1)/2)}=x_{(3)}=3 \]
Frequency is maximized at 3, so the mode equals 3.
Interpretation and diagnostic value
Mean and median separation indicates skewness in many practical datasets. A right-skewed distribution often shows \(\bar{x}>\tilde{x}\), while a left-skewed distribution often shows \(\bar{x}<\tilde{x}\). Mode location highlights the most common region and can be less stable under small samples.
- Sensitivity to outliers: mean high, median low, mode variable.
- Ordering dependence: median requires sorted data; mean does not.
- Multiplicity of modes: unimodal, bimodal, or multimodal outcomes occur under ties.
- Absence of a mode: all values distinct implies no repeating value in the usual discrete sense.
Grouped data note (frequency tables)
Frequency tables support the same concepts with weighted arithmetic. For values \(x_j\) with frequencies \(f_j\), the total count is \(n=\sum f_j\), and the weighted mean is \(\displaystyle \bar{x}=\frac{\sum f_j x_j}{\sum f_j}\). Median location depends on cumulative frequency, and mode corresponds to the highest frequency (or the modal class in grouped intervals).