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Statistics Calculator

Analyze your data set

Mean

27.00

Median

26.50

Std Dev

10.781

10 numbers entered

Central Tendency

27.00

Mean

26.50

Median

None

Mode

Dispersion

10.781

Std Dev (σ)

116.222

Variance (σ²)

33.00

Range

Summary

10

Count

270.00

Sum

12

Min

45

Max

Z-Score

Value: 25

-0.186

Z-Score

Data Distribution

1245
MeanMedian±1 Std Dev

Formulas

Mean: μ = Σx / n
Variance: σ² = Σ(x - μ)² / (n-1)
Std Dev: σ = √variance
Z-Score: z = (x - μ) / σ

Formulas Used

Mean (Average)

Mean = Σx / n

The mean is the sum of all values divided by the count of values.

Where:

Σx= Sum of all data values
n= Number of data values

Sample Standard Deviation

σ = √(Σ(x - μ)² / (n - 1))

The sample standard deviation measures data spread, using n-1 (Bessel's correction) for sample data.

Where:

x= Each data value
μ= The mean of the data
n= Number of data values

Z-Score

z = (x - μ) / σ

The Z-score indicates how many standard deviations a value is from the mean.

Where:

x= The value to evaluate
μ= The mean of the data set
σ= The standard deviation

Example Calculations

1Sample Statistics: 12, 15, 18, 22, 25, 28, 30, 35, 40, 45

Inputs

Data Set12, 15, 18, 22, 25, 28, 30, 35, 40, 45
TypeSample (n-1)

Result

Mean27.00
Median26.50
Std Dev10.781
Variance116.222
Range33.00
Count10

Sum = 270, Count = 10, Mean = 270/10 = 27.00. Sorted data median = (25+28)/2 = 26.50. No mode (each value appears once). Sample variance = 1046/9 = 116.222, Std Dev = √116.222 = 10.781.

2Small Data Set: 5, 5, 8, 10, 12

Inputs

Data Set5, 5, 8, 10, 12
TypeSample (n-1)

Result

Mean8.00
Median8.00
Mode5
Std Dev3.082
Range7.00
Count5

Sum = 40, Count = 5, Mean = 40/5 = 8.00. Sorted: 5,5,8,10,12 so Median = 8.00. Mode = 5 (appears twice). Sample variance = 38/4 = 9.500, Std Dev = √9.500 = 3.082.

Frequently Asked Questions

Q

What is the difference between mean, median, and mode?

Mean is the average (sum divided by count). Median is the middle value when data is sorted. Mode is the most frequently occurring value. Each measures central tendency differently.

  • Mean: best for symmetric distributions without outliers (e.g., test scores)
  • Median: best for skewed data or data with outliers (e.g., income, home prices)
  • Mode: best for categorical data or finding the most common value
  • For a perfectly normal distribution, mean = median = mode
  • If mean > median, the data is right-skewed; if mean < median, left-skewed
MeasureFormulaBest ForAffected by Outliers?
MeanΣx / nSymmetric dataYes — heavily
MedianMiddle valueSkewed dataNo — robust
ModeMost frequentCategorical dataNo
Q

When should I use sample vs population standard deviation?

Use sample standard deviation (n-1 divisor) when your data is a subset of a larger population. Use population standard deviation (n divisor) when you have data for the entire population.

  • Sample (n−1): surveys, experiments, clinical trials with a subset of subjects
  • Population (n): census data, all students in a class, complete inventory counts
  • Bessel’s correction (n−1) prevents underestimating true population variance
  • For n > 30, the difference between n and n−1 becomes negligible (<3%)
  • Most statistical software defaults to sample (n−1) unless specified
FeatureSample (n−1)Population (n)
Divisorn − 1n
Symbolsσ
Use whenData is a subsetYou have ALL data
Example (n=10, SS=90)s = √10 = 3.16σ = √9 = 3.00
Q

What is a Z-score?

A Z-score indicates how many standard deviations a value is from the mean. A Z-score of 0 means the value equals the mean. Positive values are above the mean, negative values are below.

  • Z = 0: the value equals the mean exactly
  • Z = ±1: within 68.3% of data in a normal distribution
  • Z = ±2: within 95.4% of data — values beyond are unusual
  • Z = ±3: within 99.7% of data — values beyond are rare outliers
  • Z-scores allow comparing values from different data sets (e.g., SAT vs ACT)
Q

What is variance and how does it relate to standard deviation?

Variance measures how spread out data is from the mean. It is the average of squared differences from the mean. Standard deviation is the square root of variance and is in the same units as your data.

  • Variance = Σ(x − μ)² / n (population) or / (n−1) (sample)
  • Standard deviation = √Variance, reported in the same units as the data
  • Variance is always non-negative; zero means all values are identical
  • Doubling all values quadruples the variance but only doubles the std dev
  • Coefficient of variation (CV) = std dev / mean × 100% for relative comparison

Understanding Descriptive Statistics

Descriptive statistics summarize and describe the main features of a data set. They provide simple summaries about the sample and the measures.

Measures of central tendency (mean, median, mode) describe the center of a distribution. Measures of dispersion (range, variance, standard deviation) describe how spread out the data is.

Z-scores standardize data, allowing comparison between different data sets or determining how unusual a particular value is within a distribution.

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Last Updated: Mar 26, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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