Statistics Calculator

Mean is the arithmetic average (sum divided by count). Median is the middle value when data is sorted. Median is better for skewed data with outliers. Example: incomes of [30k, 35k, 40k, 45k, 1M] - mean is $230k (misleading), median is $40k (more representative).

Standard deviation is in original units, easier to interpret. Variance (squared units) is used in mathematical calculations like ANOVA. Example: heights measured in inches - standard deviation is in inches (intuitive), variance is in square inches (harder to understand). Use standard deviation for reporting, variance for statistical tests.

Data points are spread far from the mean (high variability). Low standard deviation means data clusters near the average (consistency). Example: two classes both average 80% on tests. Class A: std dev = 5 (most scores 75-85, consistent). Class B: std dev = 15 (scores range 50-100, wide variation in student performance).

Mean and standard deviation are sensitive to outliers. Median and mode are resistant. Consider using median for skewed data. One extreme outlier can dramatically change the mean while barely affecting the median. Always check for outliers and consider whether to include them based on whether they represent genuine data or errors.

A z-score tells you how many standard deviations a value is from the mean. Z = 0 is the mean, Z = 1 is one standard deviation above, Z = -2 is two standard deviations below. Z-scores allow comparison of values from different datasets. Example: height z-score of +1.5 means taller than 93% of the population.

Values dividing sorted data into four equal parts. Q1 (25th percentile), Q2/Median (50th percentile), Q3 (75th percentile). IQR = Q3 - Q1 measures middle spread. Quartiles are useful for understanding data distribution and identifying outliers. Box plots visualize quartiles clearly.

Values beyond 1.5xIQR from Q1 or Q3 are mild outliers. Beyond 3xIQR or 3 standard deviations from mean are extreme outliers. Outliers may represent errors, rare events, or genuine extreme values. Investigate outliers before deciding to remove them. Sometimes the outlier is the most interesting data point.

If all values appear equally often, there is no mode. Some datasets have multiple modes (bimodal, trimodal). The mode is most useful for categorical data or discrete numerical data with repeated values. For continuous data (like exact measurements), the mode may not be meaningful.

Population: divide by N. Sample: divide by n-1 (Bessel's correction). Use sample when data is a subset of the target group. The n-1 correction makes the sample variance an unbiased estimator of the population variance. For large samples (n greater than 30), the difference is minimal.

Bell-shaped curve where data clusters symmetrically around the mean. Many natural phenomena follow this pattern: heights, test scores, measurement errors. In normal distribution, mean = median = mode. The 68-95-99.7 rule applies only to normal distributions. Not all data is normally distributed - income, for example, is usually right-skewed.

At least 5-10 for basic descriptive stats. 30+ for inferential statistics and significance testing. More data generally gives more reliable results. However, quality matters too - 10 accurate measurements may be better than 100 sloppy ones. For highly variable data, you need more points to get stable estimates.

Descriptive summarizes data (mean, std dev) - describes what you have. Inferential draws conclusions about populations from samples - estimates what you do not know. This calculator provides descriptive statistics. For inferential stats (confidence intervals, hypothesis tests), you need additional tools.

Use mode, frequency counts, and proportions for categorical data. Mean and standard deviation require numerical data. For yes/no or category data, count frequencies and calculate percentages. Chi-square tests can analyze relationships between categorical variables.

Standard deviation divided by mean, expressed as percentage. Allows comparing variability across different scales. Example: comparing consistency of two athletes - one runs 100m (std dev 0.5s, mean 12s, CV = 4.2%), one runs marathon (std dev 5min, mean 4hr, CV = 2.1%). The marathon runner is actually more consistent relative to their event distance.

Uses precise floating-point math with 15+ significant digits. Results rounded for display but internally precise. Suitable for educational and most professional purposes. For critical applications requiring certified accuracy, use specialized statistical software. Round-off errors can accumulate in extreme cases with very large datasets or numbers.