Summary Statistics Calculator

Mean · median · mode · SD · quartiles · histogram

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What Are Summary Statistics?

Summary statistics (also called descriptive statistics) condense a dataset into a handful of meaningful numbers. Instead of looking at hundreds of raw values, you get three key aspects: where the data centers, how spread out it is, and what shape the distribution takes.

Analysis Mean = 85.67 SD = 5.46 Q1, Q2, Q3 Skew = −0.12
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Central Tendency

Mean, median, and mode tell you where the “center” of your data lies — the typical or representative value.

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Dispersion

Range, variance, standard deviation, and IQR measure how spread out values are around the center.

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Distribution Shape

Skewness and kurtosis reveal whether data is symmetric, skewed, or has heavy/light tails.

Measures of Central Tendency

Mean:  x̄ = Σxi / n

The arithmetic mean sums all values and divides by the count. It uses every data point, making it sensitive to outliers.

Median:  Middle value when data is sorted

The median is the middle value of sorted data. For even n, it averages the two middle values. Resistant to outliers.

Mode:  Most frequently occurring value(s)

The mode is the value that appears most often. Data can be unimodal, bimodal, multimodal, or have no mode.

When to Use Each Measure

MeasureBest ForLimitation
MeanSymmetric data, no outliersDistorted by extreme values
MedianSkewed data, outliers presentIgnores actual values of extremes
ModeCategorical data, finding peaksMay not exist or be unique

Measures of Dispersion

Variance:  s² = Σ(xi − x̄)² / (n − 1)
Standard Deviation:  s = √s²
Range:  max − min    |    IQR:  Q3 − Q1

Worked Example

Data: [2, 4, 4, 4, 5, 5, 7, 9]
Step 1: Mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
Step 2: Deviations from mean: −3, −1, −1, −1, 0, 0, 2, 4
Step 3: Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
Step 4: Sum of squares = 32
Step 5: Variance (sample) = 32/(8−1) = 4.5714
Step 6: SD = √4.5714 = 2.1381

The 68–95–99.7 Rule (Empirical Rule)

For normally distributed data:

μ −1σ +1σ −2σ +2σ 68.3% 95.4%

Understanding Distribution Shape

Skewness

Left-Skewed skew < 0 Symmetric skew ≈ 0 Right-Skewed skew > 0
|Skewness|Interpretation
< 0.5Approximately symmetric
0.5 – 1.0Moderately skewed
> 1.0Highly skewed

Kurtosis

Mesokurtic (k ≈ 0)

Normal distribution shape. Tails contain roughly the expected proportion of data.

Leptokurtic (k > 0)

Peaked with heavy tails. More extreme values (outliers) than normal. Example: stock returns.

Platykurtic (k < 0)

Flat with light tails. Fewer extreme values than normal. Example: uniform-like data.

Quartiles & Box Plots

Min Q1 Q2 (Median) Q3 Max Outlier IQR = Q3 − Q1
Five-Number Summary:  Min, Q1, Median, Q3, Max
Outlier Detection:  value < Q1 − 1.5×IQR  or  value > Q3 + 1.5×IQR

The box plot (box-and-whisker) visualizes the five-number summary. The box spans Q1 to Q3 (the middle 50% of data), the line inside marks the median, and whiskers extend to the most extreme non-outlier values. Points beyond the fences appear as individual dots.

Frequently Asked Questions

Sample standard deviation divides by n−1 (Bessel’s correction) because a sample underestimates variability. Population standard deviation divides by n because you have all data points. Use sample SD (s) when analyzing a subset of a larger population; use population SD (σ) when you have the entire dataset. Our calculator shows both.
Skewness measures asymmetry: 0 means symmetric, positive means right-skewed (long right tail), negative means left-skewed. Kurtosis (excess) measures tail weight: 0 is normal, positive means heavier tails (leptokurtic), negative means lighter tails (platykurtic). Values between −0.5 and 0.5 are approximately symmetric or normal.
Quartiles divide sorted data into four equal parts. Q1 (25th percentile) has 25% of data below it. Q2 is the median. Q3 (75th percentile) has 75% below. The IQR (Q3 − Q1) measures the spread of the middle 50% and is used for outlier detection: any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is flagged.
Use the mean for symmetric data without outliers — it uses all values efficiently. Use the median for skewed data or data with outliers — it is resistant to extreme values. For example, median income is preferred over mean income because a few billionaires skew the mean upward significantly.
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It measures relative variability, allowing you to compare spread between datasets with different units or scales. A CV below 15% generally indicates low variability; above 30% indicates high variability.
The IQR method flags values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR as outliers (the same rule used in box plots). The z-score method flags values more than 2 or 3 standard deviations from the mean. Always investigate outliers before removing them — they may represent real phenomena or data entry errors.

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