Descriptive Statistics and Probability: Measures of Variation, Probability Concepts, and Counting Rules

Measures of Variation

Range

The range is a simple measure of variation that indicates the spread between the largest and smallest values in a data set.

• Definition: The difference between the maximum and minimum data entries in the set.

• The data must be quantitative.

• Formula:

$\text{Range} = (\text{Max. data entry}) - (\text{Min. data entry})$

Variation

Variation describes how data values are spread out or clustered together. Two data sets can have the same mean but different variations.

• Greater variation means data values are more spread out.

• Example: Two corporations with the same mean starting salary but different spreads (see bar charts for Corporation A and B).

Deviation, Variance, and Standard Deviation

• Deviation: The difference between a data entry and the mean of the data set.

• Population deviation: $x - \mu$

• Sample deviation: $x - \bar{x}$

• Population Variance:

$\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

• Population Standard Deviation:

$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

• Sample Variance:

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

• Sample Standard Deviation:

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

• Observations:

• Standard deviation measures the spread of data around the mean.

• Standard deviation is always non-negative; it is zero only if all entries are identical.

• Larger standard deviation means data are more spread out.

Step-by-Step Calculation: Population Variance & Standard Deviation

Step	In Words	In Symbols
1	Find the mean of the population data set	$\mu = \frac{\sum x}{N}$
2	Find deviation of each entry	$x - \mu$
3	Square each deviation	$(x - \mu)^2$
4	Add to get the sum of squares	$SS_x = \sum (x - \mu)^2$
5	Divide by N to get the population variance	$\sigma^2 = \frac{\sum (x - \mu)^2}{N}$
6	Find the square root to get the population standard deviation	$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

Step-by-Step Calculation: Sample Variance & Standard Deviation

Step	In Words	In Symbols
1	Find the mean of the sample data set	$\bar{x} = \frac{\sum x}{n}$
2	Find deviation of each entry	$x - \bar{x}$
3	Square each deviation	$(x - \bar{x})^2$
4	Add to get the sum of squares	$SS_x = \sum (x - \bar{x})^2$
5	Divide by n - 1 to get the sample variance	$s^2 = \frac{\sum (x - \bar{x})^2}{n-1}$
6	Find the square root to get the sample standard deviation	$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$

Interpreting Standard Deviation

• Standard deviation measures the typical amount an entry deviates from the mean.

• Greater spread in data means a larger standard deviation.

Empirical Rule (68-95-99.7 Rule)

For data with a symmetric, bell-shaped distribution:

• About 68% of data lie within one standard deviation of the mean.

• About 95% within two standard deviations.

• About 99.7% within three standard deviations.

Chebyshev's Theorem

• For any data set, the proportion of values within k standard deviations (k > 1) of the mean is at least $1 - \frac{1}{k^2}$.

• For k = 2: at least 75% of data within 2 standard deviations.

• For k = 3: at least 88.9% of data within 3 standard deviations.

Standard Deviation for Grouped Data

• For frequency distributions, use class midpoints and frequencies:

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

• Where f = frequency of each class.

Coefficient of Variation (CV)

• Describes the standard deviation as a percent of the mean.

Population data set: $CV = \frac{\sigma}{\mu} \times 100\%$

Sample data set: $CV = \frac{s}{\bar{x}} \times 100\%$

Quartiles, Interquartile Range, and Boxplots

Quartiles

• Quartiles divide an ordered data set into four equal parts.

• Q1: About 25% of data fall on or below Q1.

• Q2: Median; about 50% of data fall on or below Q2.

• Q3: About 75% of data fall on or below Q3.

Interquartile Range (IQR)

• Measures the range of the middle 50% of the data.

• Formula: $IQR = Q3 - Q1$

Using IQR to Identify Outliers

1. Find Q1 and Q3.

2. Compute IQR: $IQR = Q3 - Q1$

3. Multiply IQR by 1.5: $1.5 \times IQR$

4. Subtract $1.5 \times IQR$ from Q1. Data below this are outliers.

5. Add $1.5 \times IQR$ to Q3. Data above this are outliers.

Box and Whisker Plot

• Exploratory data analysis tool that highlights important features of a data set.

• Requires the five-number summary:

  1. Minimum entry

  2. First quartile (Q1)

  3. Median (Q2)

  4. Third quartile (Q3)

  5. Maximum entry

Steps to Draw a Box and Whisker Plot

1. Find the five-number summary.

2. Construct a horizontal scale for the data range.

3. Plot the five numbers above the scale.

4. Draw a box from Q1 to Q3, with a line at Q2 (median).

5. Draw whiskers from the box to the minimum and maximum entries.

Percentiles and Other Fractiles

Fractiles	Summary	Symbols
Quartiles	Divides data into 4 equal parts	Q1, Q2, Q3
Deciles	Divides data into 10 equal parts	D1, D2, D3, ..., D9
Percentiles	Divides data into 100 equal parts	P1, P2, ..., P99

Percentile of a Data Entry

• To find the percentile that corresponds to a specific data entry x:

Percentile of x = $\frac{\text{number of data entries less than x}}{\text{total number of data entries}} \times 100$

The Standard Score (z-score)

• Represents the number of standard deviations a value x falls from the mean μ.

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

Probability: Basic Concepts and Counting

Probability Experiments

• Probability experiment: An action or trial with specific results (counts, measurements, or responses).

• Outcome: The result of a single trial.

• Sample space: The set of all possible outcomes.

• Event: One or more outcomes; a subset of the sample space.

Simple and Compound Events

• Simple event: Consists of a single outcome (e.g., tossing heads and rolling a 3).

• Compound event: Consists of more than one outcome (e.g., tossing heads and rolling an even number).

The Fundamental Counting Principle

• If one event can occur in m ways and a second in n ways, the two events can occur in $m \times n$ ways.

• Can be extended for more events in sequence.

Types of Probability

• Classical (theoretical) probability: Each outcome is equally likely.

$P(E) = \frac{\text{Number of outcomes in event E}}{\text{Number of outcomes in sample space}}$

• Empirical (statistical) probability: Based on observed data.

$P(E) = \frac{f}{n}$, where f = frequency of event E, n = total frequency

• Subjective probability: Based on intuition, educated guesses, or estimates.

Law of Large Numbers

• As an experiment is repeated, the empirical probability approaches the theoretical probability.

• Example: Probability of tossing a head approaches 0.5 as the number of tosses increases.

Range of Probabilities Rule

• Probability of any event E is between 0 and 1, inclusive: $0 \leq P(E) \leq 1$

Complementary Events

• The complement of event E (denoted E') is the set of all outcomes not in E.

• $P(E) + P(E') = 1$

• $P(E') = 1 - P(E)$

Conditional Probability and the Multiplication Rule

Conditional Probability

• The probability of event B occurring, given that event A has already occurred.

• Denoted $P(B|A)$ (read as "probability of B, given A").

Independent and Dependent Events

• Independent events: The occurrence of one does not affect the probability of the other.

• $P(B|A) = P(B)$ or $P(A|B) = P(A)$

• Events that are not independent are dependent.

The Multiplication Rule

• For two events A and B, the probability that both occur in sequence:

• General rule: $P(A \text{ and } B) = P(A) \cdot P(B|A)$

• For independent events: $P(A \text{ and } B) = P(A) \cdot P(B)$

• Can be extended for more than two events.