Fundamental Concepts and Formulas in Statistics

Descriptive Statistics

Class Width

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

• Formula:

$\text{Class width} = \frac{\text{max value} - \text{min value}}{\text{number of classes}}$

Sample Mean

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

• Formula (raw data):

$\bar{x} = \frac{\sum x_i}{n}$

• Formula (frequency table):

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$

Sample Standard Deviation

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

• Formula (definition):

$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$

• Formula (shortcut):

$s = \sqrt{\frac{\sum x^2 - (\sum x)^2 / n}{n-1}}$

Midrange

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

• Formula:

$\text{midrange} = \frac{\text{max} + \text{min}}{2}$

Probability and Counting Rules

Permutation

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

• Formula (n objects, r at a time):

$nPr = \frac{n!}{(n-r)!}$

• Formula (with identical items):

$n! / (n_1! n_2! ... n_k!)$

Combination

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

• Formula:

$nCr = \frac{n!}{r!(n-r)!}$

Conditional Probability

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

• Formula:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Addition Rule

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

• Formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Measures of Position and Spread

Z-score

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

• Formula:

$z = \frac{x - \bar{x}}{s}$

$z = \frac{x - \mu}{\sigma}$

IQR and Upper and Lower Fences

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

• Formulas:

$\text{IQR} = Q_3 - Q_1$

$LF = Q_1 - 1.5(\text{IQR})$

$UF = Q_3 + 1.5(\text{IQR})$

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.

Standard Deviations from Mean	Percent of Data
1	68%
2	95%
3	99.7%