20
—
20
50
62.5
7.0711
7.9057
20
—
20
50
62.5
7.0711
7.9057
The Statistics Calculator computes essential descriptive statistics for a dataset of five values. Descriptive statistics summarize data through measures of central tendency and dispersion, providing a concise picture of the dataset's characteristics.
The mean (arithmetic average) is the most common measure of center:
$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}$$
The median is the middle value when data are sorted in ascending order. For an odd number of values $$n$$, the median is the value at position $$(n+1)/2$$. The median is more robust to outliers than the mean—a single extreme value can dramatically shift the mean but leaves the median nearly unchanged.
The range is the simplest measure of spread: $$\text{Range} = x_{\max} - x_{\min}$$. While easy to compute, it is highly sensitive to outliers.
The variance measures the average squared deviation from the mean. The population variance divides by $$n$$, while the sample variance uses Bessel's correction (dividing by $$n-1$$) to produce an unbiased estimate:
$$\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2, \qquad s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$$
The standard deviation is the square root of the variance, expressed in the same units as the data:
$$\sigma = \sqrt{\sigma^2}, \qquad s = \sqrt{s^2}$$
If your five values represent an entire population, use the population variance ($$\sigma^2$$, dividing by $$n = 5$$). If they are a sample drawn from a larger population, use the sample variance ($$s^2$$, dividing by $$n - 1 = 4$$). The sample variance is unbiased—on average, it correctly estimates the true population variance.
Descriptive statistics are the foundation of data analysis in every field. In quality control, they monitor manufacturing processes. In medicine, they summarize clinical trial results. In finance, the mean return and standard deviation characterize investment performance. In education, they describe test score distributions. Understanding these basic measures is essential before proceeding to any inferential statistical analysis.
Enter five numerical values. The calculator computes the mean, finds the median by sorting, calculates the range, and determines both population and sample variance and standard deviation. All results update instantly as you change any input.
A larger standard deviation indicates greater spread in the data. If the mean and median are close, the data is roughly symmetric; if they differ substantially, the data may be skewed. Use the sample statistics when your data represents a sample from a larger group, and population statistics when you have the complete dataset.
Inputs
Results
The mean is 86.6 and median is 88. The small gap suggests slight left skew. The sample standard deviation of 5.32 indicates moderate score variation.
Inputs
Results
Mean equals median at 21, indicating a symmetric distribution. The standard deviation of 1.41°C shows the readings are tightly clustered.
Use population standard deviation ($$\sigma$$) when your data includes every member of the group you're studying. Use sample standard deviation ($$s$$) when your data is a subset drawn from a larger population and you want to estimate the population parameter. The sample version divides by $$n-1$$ instead of $$n$$ to correct for bias.
The median is the middle value in a sorted dataset. For 5 values, it is the 3rd smallest. The median is resistant to outliers: if one value is extremely large or small, the median remains stable while the mean shifts dramatically. This makes it preferred for skewed distributions like income data.
Variance measures how spread out the data values are from the mean. Mathematically, it is the average of the squared deviations from the mean. A variance of zero means all values are identical. Larger variance indicates more dispersion.
Standard deviation is in the same units as the original data, making it directly interpretable. If you measure heights in centimeters, the standard deviation is also in centimeters, while the variance is in centimeters squared—a less intuitive unit.
This calculator is designed for exactly 5 values to provide instant results. For larger datasets, you would use the same formulas with a larger $$n$$. The mean formula generalizes naturally, and the median would be the middle value (or average of two middle values for even $$n$$).
Bessel's correction is dividing by $$n-1$$ instead of $$n$$ when computing sample variance. This compensates for the fact that a sample tends to underestimate the true population spread (since the sample mean is closer to the sample values than the population mean is). The correction makes the sample variance an unbiased estimator of the population variance.
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