Statistics Calculator

A mean score is an average score, often denoted by X. It is the sum of all the scores in a data set divided by the number of observations in the data set. Thus, if you have a set of N numbers ( X₁ , X₁ , X₁ , . . . X_N ), the mean of those numbers would be:

X = ( X₁ + X₂ + X₃ + . . . + X_N ) / N = [ Σ X_i ] / N

For example, the mean of the numbers 1, 2, and 3 would be (1 + 2 + 3)/3 or 2.

The median is a simple measure of central tendency. To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

Thus, in a sample of four families, we might want to compute the median annual income. Suppose the incomes are $30,000 for the first family; $50,000, for the second; $90,000, for the third; and $110,000, for the fourth. The two middle values are $50,000 and $90,000. Therefore, the median annual income is ($50,000 + $90,000)/2 or $70,000.

The mode is the most frequently appearing value in a population or sample.

Suppose we draw a sample of five boys and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds. Since more boys weigh 100 pounds than any other weight, the sample mode would equal 100 pounds.

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the standard deviation of a population and the standard deviation of a sample. They have different notation, and they are computed differently. The standard deviation of a population is denoted by σ and the standard deviation of a sample, by s.

The standard deviation of a population is defined by the following formula:

σ = sqrt [ Σ ( X_i - μ)² / N ]

where σ is the population standard deviation, μ is the population mean, X_i is the ith element from the population, and N is the number of elements in the population.

The standard deviation of a sample is defined by slightly different formula:

s = sqrt [ Σ ( x_i - x)² / (n - 1) ]

where s is the sample standard deviation, x is the sample mean, x_i is the ith element from the sample, and n is the number of elements in the sample.

Note: The standard deviation is the square root of the variance.

The variance is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the variance of a population and the variance of a sample. They have different notation, and they are computed differently. The variance of a population is denoted by σ² and the variance of a sample, by s².

The variance of a population is defined by the following formula:

σ² = Σ ( X_i - μ)² / N

where σ² is the population variance, μ is the population mean, X_i is the ith element from the population, and N is the number of elements in the population.

The variance of a sample is defined by slightly different formula:

s² = Σ ( x_i - x)² / (n - 1)

where s² is the sample variance, x is the sample mean, x_i is the ith element from the sample, and n is the number of elements in the sample.

Note: The square root of the variance is the standard deviation.

The range is a simple measure of variation in a data set. It is the difference between the biggest and smallest observation in the data set.

Range = Maximum value - Minimum value

For example, suppose a data set consisted of the following five observations: (3, 5, 5, 7, 8}. The range in that data set would be 8 - 3 or 5.

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

• Q2 is the median of the data set.
• Q1 is the "middle" value in the first half of the rank-ordered data set (i.e., the values that come before the median).
• Q3 is the "middle" value in the second half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider a data set made up of the following nine numbers: 1, 3, 4, 4, 5, 6, 6, 7, 11.

• Q2 is the median of the rank-ordered data set (i.e., the middle value). In this example, the middle value is 5.
• Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5.
• Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5.
• The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

• Q2 is the median of the data set.
• Q1 is the "middle" value in the first half of the rank-ordered data set (i.e., the values that come before the median).
• Q3 is the "middle" value in the second half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider a data set made up of the following nine numbers: 1, 3, 4, 4, 5, 6, 6, 7, 11.

• Q2 is the median of the rank-ordered data set (i.e., the middle value). In this example, the middle value is 5.
• Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5.
• Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5.
• The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

• Q2 is the median of the data set.
• Q1 is the "middle" value in the first half of the rank-ordered data set (i.e., the values that come before the median).
• Q3 is the "middle" value in the second half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider a data set made up of the following nine numbers: 1, 3, 4, 4, 5, 6, 6, 7, 11.

• Q2 is the median of the rank-ordered data set (i.e., the middle value). In this example, the middle value is 5.
• Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5.
• Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5.
• The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

Data sets with fewer observations on the right (toward higher values) are said to be skewed right; and data sets with fewer observations on the left (toward lower values) are said to be skewed left.

Skewed right Skewed left

Skewness is a quantitative measure of the degree to which a data set skews left or skews right. Here is a formula for computing skewness of sample data:

where Skew_s is sample skewness, x_i is value for observation i, x is the sample mean, s is the sample standard deviation, and n is the number of observations in the data set.

Here is a formula for computing skewness of population data:

where Skew_p is population skewness, x_i is value for observation i, μ is the population mean, σ is the population standard deviation, and N is the number of observations in the population.

Data sets that are roughly symmetric have a skewness value near zero. Data sets that skew left have a negative skewness; data sets that skew right have a positive skewness.

Note: There are different ways to measure skewness. This calculator uses the same formula as SPSS and Excel. But, if you use different software, you may get a different result.

Certain computations require a minimum number of observations. When the number of observations in a data set is insufficient, the calculator displays "NaN" as an output.

Here are the requirements for outputs that require a minimum number of observations:

• The sample variance and the sample standard deviation each require at least two observations.
• Population skewness requires at least two observations.
• Sample skewness requires at least three observations.
• The lower quartile (Q1), the upper quartile (Q3), and the interquartile range require at least three observations.