Descriptive Statistics

Class Width

The class width is used in frequency distributions to determine the range of each class interval.

• Formula:

Sample Mean

The sample mean is the average value of a sample, representing the central tendency.

• Formula (raw data):

• Formula (frequency table):

Sample Standard Deviation

The sample standard deviation measures the spread or dispersion of data points in a sample.

• Formula (definition):

• Formula (shortcut):

Midrange

The midrange is a measure of central tendency, calculated as the average of the maximum and minimum values.

• Formula:

Probability and Counting Rules

Permutation

Permutations count the number of ways to arrange items where order matters.

• Formula (n objects, r at a time):

• Formula (with identical items):

Combination

Combinations count the number of ways to select items where order does not matter.

• Formula:

Conditional Probability

Conditional probability is the probability of event A occurring given that event B has occurred.

• Formula:

Addition Rule

The addition rule calculates the probability of either event A or event B occurring.

• Formula:

Measures of Position and Spread

Z-score

The z-score indicates how many standard deviations a data point is from the mean.

• Formula:

IQR and Upper and Lower Fences

The interquartile range (IQR) measures the spread of the middle 50% of data. Fences help identify outliers.

• Formulas:

Empirical Rule

Empirical Rule (68-95-99.7 Rule)

The empirical rule describes the spread of data in a normal distribution.

• Approximately 68% of the data is within 1 standard deviation of the mean.

• Approximately 95% of the data is within 2 standard deviations of the mean.

• Approximately 99.7% of the data is within 3 standard deviations of the mean.

Example: If the mean test score is 70 and the standard deviation is 10, about 68% of scores will fall between 60 and 80.