Standard Deviation Calculator With Outlier Tests

Choose Sample or Population Standard Deviation

Use the population convention to describe the complete set you entered. Use the sample convention when the entries are a sample of a wider population.

Population variance divides the sum of squared deviations by the count, n. Sample variance divides by n − 1. Standard deviation is the square root of variance, expressed in the original amount units.

The default amounts are 2, 4, 4, 4, 5, 5, 7 and 9. Their population variance is 4 and population standard deviation is 2. Sample variance is 32/7, giving a sample standard deviation of about 2.14.

The export records the selected convention and both results. Blank and invalid entries retain their entry numbers and exclusion status. They never become zero. Every delimiter starts another entry, so a blank line also counts as an excluded entry.

Calculate Z Scores and Outliers

A z score divides an amount’s distance from the mean by the selected standard deviation: z = (amount − mean) / standard deviation. Use a nonzero standard deviation.

The row and comparison scores retain your sample or population convention. The z-score test flags a row when its absolute score exceeds your cutoff.

The IQR test uses the middle half of the sorted amounts. This worksheet calculates quartiles by linear interpolation at position (n − 1) × p, with zero-based positions. This is the type 7 convention.

With the default values, Q1 is 4 and Q3 is 5.5. An IQR multiplier of 1.5 gives fences of 1.75 and 7.75, so the amount 9 is flagged.

Using population standard deviation, 9 has a z score of exactly 2. It is within a cutoff of 2 because the rule uses a strict comparison. This explains the different flag sets.

The z-score-to-percentile calculator converts an entered z score using the standard normal cumulative distribution. Its model percentile is the percentage of that distribution at or below the score.

For z = 1.96, the model percentile is 97.5%. A score of −1.96 gives 2.5%, and zero gives 50%.

This conversion assumes a normal distribution with mean zero and standard deviation one. It works independently of the entered amounts and accepts z scores from −8 to 8.

An empirical rank counts observations in your actual dataset. The model percentile does not calculate that rank or establish that transaction amounts follow a normal distribution.

A comparison amount gets a score against the entered dataset without changing its mean or spread. One entry has no sample standard deviation. Identical amounts have zero spread, so their z scores are unavailable.

Follow the Calculation Steps

The default amounts total 40 across eight entries, giving a mean of 5. Subtracting that mean produces deviations of −3, −1, −1, −1, 0, 0, 2 and 4.

Squaring those deviations gives 9, 1, 1, 1, 0, 0, 4 and 16. Their sum is 32. Divide by eight for population variance or seven for sample variance, then take the square root.

Each row shows its deviation and squared deviation so you can reproduce the calculation. Displayed values use up to two decimal places. Exports retain the underlying numerical precision.

Decimal amounts and scientific notation are accepted. Write a fraction such as one-half as 0.5. The tool calculates its own mean and does not pool separate groups or accept a supplied mean.

The NIST explanation of measures of scale describes sample variance, standard deviation and the IQR. Its tail-behaviour examples explain why extreme amounts can strongly affect standard deviation.

Interpret a Flagged Transaction

A statistical flag identifies an amount outside the chosen rule. It does not establish fraud. Skewed transaction amounts can produce legitimate flags and can also increase standard deviation enough to conceal unusual rows.

Separate currencies and materially different transaction types before comparing amounts. Mixing wages with asset sales creates a population whose spread can tell you little about either group.

Check the original statement and supporting invoice for a flagged entry. Confirm the amount and date, then examine its counterparty and purpose. Check duplicate entries and reversals before drawing a conclusion.

Use forensic data analytics to place this screen within a wider examination of transaction records.

Screen First-Digit Frequencies

Turn on the optional worksheet to compare observed first-digit counts with Benford expected counts. For digit d, the expected count is N × log10(1 + 1/d), where N is the included count.

Zero has no first significant digit and is excluded. Positive-only mode also excludes negative amounts. Absolute-value mode includes their magnitudes: −250 contributes to digit 2 and 0.004 contributes to digit 4.

For 100 included values, digit 1 has an expected count of about 30.1 and digit 9 about 4.58. Expected counts need not be whole numbers.

Assigned identifiers and amounts constrained to a narrow range are unsuitable for a simple Benford comparison. Small datasets give unstable frequency comparisons because each observation changes a large share of the count. The worksheet provides counts without a fraud verdict or statistical significance test.

Read when Benford’s law is useful before applying the comparison. A difference is an investigation lead that needs source-record checks.