In statistics, a z-score tells us how many standard deviations away a value is from the mean. We use the following formula to calculate a z-score:
z = (X – μ) / σ
where:
- X is a single raw data value
- μ is the population mean
- σ is the population standard deviation
This tutorial explains how to calculate z-scores for raw data values in R.
Example 1: Find Z-Scores for a Single Vector
The following code shows how to find the z-score for every raw data value in a vector:
#create vector of data data <- c(6, 7, 7, 12, 13, 13, 15, 16, 19, 22) #find z-score for each data value z_scores <- (data-mean(data))/sd(data) #display z-scores z_scores [1] -1.3228757 -1.1338934 -1.1338934 -0.1889822 0.0000000 0.0000000 [7] 0.3779645 0.5669467 1.1338934 1.7008401
Each z-score tells us how many standard deviations away an individual value is from the mean. For example:
- The first raw data value of “6” is 1.323 standard deviations below the mean.
- The fifth raw data value of “13” is 0 standard deviations away from the mean, i.e. it is equal to the mean.
- The last raw data value of “22” is 1.701 standard deviations above the mean.
Example 2: Find Z-Scores for a Single Column in a DataFrame
The following code shows how to find the z-score for every raw data value in a single column of a dataframe:
#create dataframe df <- data.frame(assists = c(4, 4, 6, 7, 9, 13), points = c(24, 29, 13, 15, 19, 22), rebounds = c(5, 5, 7, 8, 14, 15)) #find z-score for each data value in the 'points' column z_scores <- (df$points-mean(df$points))/sd(df$points) #display z-scores z_scores [1] 0.6191904 1.4635409 -1.2383807 -0.9006405 -0.2251601 0.2814502
Each z-score tells us how many standard deviations away an individual value is from the mean. For example:
- The first raw data value of “24” is 0.619 standard deviations above the mean.
- The second raw data value of “29” is 1.464 standard deviations above the mean.
- The third raw data value of “13” is 1.238 standard deviations below the mean.
And so on.
Example 3: Find Z-Scores for Every Column in a DataFrame
The following code shows how to find the z-score for every raw data value in every column of a dataframe using the sapply() function.
#create dataframe df <- data.frame(assists = c(4, 4, 6, 7, 9, 13), points = c(24, 29, 13, 15, 19, 22), rebounds = c(5, 5, 7, 8, 14, 15)) #find z-scores of each column sapply(df, function(df) (df-mean(df))/sd(df)) assists points rebounds [1,] -0.92315712 0.6191904 -0.9035079 [2,] -0.92315712 1.4635409 -0.9035079 [3,] -0.34011052 -1.2383807 -0.4517540 [4,] -0.04858722 -0.9006405 -0.2258770 [5,] 0.53445939 -0.2251601 1.1293849 [6,] 1.70055260 0.2814502 1.3552619
The z-scores for each individual value are shown relative to the column they’re in. For example:
- The first value of “4” in the first column is 0.923 standard deviations below the mean value of its column.
- The first value of “24” in the second column is .619 standard deviations above the mean value of its column.
- The first value of “9” in the third column is .904 standard deviations below the mean value of its column.
And so on.
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