Statistics Calculator

Calculate mean, median, mode, standard deviation, and more

Enter your data (comma or space separated)

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

Count

270.00

Sum

Min

Max

Z-Score Calculator

Value

Z-Score

-0.186

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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

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.

Statistics Calculator

Central Tendency

Dispersion

Summary

Z-Score Calculator

Data Distribution

Complete Statistics Summary

Formulas

Formulas Used

Mean (Average)

Sample Standard Deviation

Z-Score

Example Calculations

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

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

Frequently Asked Questions

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

When should I use sample vs population standard deviation?

What is a Z-score?

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

Understanding Descriptive Statistics

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Central Tendency

Dispersion

Summary

Z-Score Calculator

Data Distribution

Complete Statistics Summary

Formulas

Formulas Used

Mean (Average)

Sample Standard Deviation

Z-Score

Example Calculations

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

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

Frequently Asked Questions

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

When should I use sample vs population standard deviation?

What is a Z-score?

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

Understanding Descriptive Statistics

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