Five Number Distribution
When a summary of the statistics is asked for, use the five number distribution. This includes the minimum, Q1, median, Q3, and maximum.
Standard Deviation and Variance
Sx2 = ((x1-x̄)2) + (x1-x̄)2) / (n-1) Sx = √(Sx2)
Sx is the standard deviation of the sample. σx is the standard deviation of the population. It is important to keep in mind that Sx measures spread about the mean and should only be used when the mean (not median) is closer as the measure of center. Remember, you use the mean when there are no outliers or strong skew visible in the distribution. For this reason, use x̄ and Sx only for reasonably symmetric distributions. Sx, like the mean, is not resistant to outliers. A couple outliers can make Sx very large.
Standardization
Standardizing is converting observation from original values to standard deviation units.
Percentile: The pth percentile of a distribution is the value with p percent of the observations less than it. Remember: a value isn’t “in” the 85th percentile; it is “at” the 85th percentile.
For example, if someone is at the 81st percentile on a test, they did better than 81% of test takers.
To standardize a value, find the z-score using this formula: Z = (x-mean)/(standard deviation)
Z-score units are not measured in the same units as the variable. It measures standard deviations from the mean.
Z-scores are directional. If the z-score is positive it means the observation is larger than the mean. If the z-score is negative, it means the observation is smaller than the mean.
For example, a z-score of 1.5 means that the data point is 1.5 standard deviations above the mean.
Transforming the entire data set
Adding the same positive number a to (or subtracting a from) each observation
- Adds a to (or subtracts a from) measures of center and location (mean, median, quartiles, percentiles)
- Doesn’t change the shape of the distribution or measures of spread (range, IQR, standard deviation)
Multiplying (or dividing) each observation by the same positive number b
- Multiplies (or divides) measures of center and location (mean, median, quartiles, percentiles), by b
- Multiplies (or divides) measures of spread (range, IQR, standard deviation) by b
- Does not change the shape of the distribution