Introductory Statistics: Exam 1 Study Guide

Levels of Measurement

Types of Measurement Scales

Understanding the levels of measurement is essential for selecting appropriate statistical methods and interpreting data correctly.

• Nominal: Data are categorized without a natural order or ranking (e.g., gender, colors).

• Ordinal: Data are categorized with a meaningful order, but differences between values are not meaningful (e.g., class rankings, satisfaction ratings).

• Interval: Ordered data with meaningful differences between values, but no true zero point (e.g., temperature in Celsius).

• Ratio: Like interval, but with a true zero point, allowing for ratios (e.g., height, weight, age).

Example: Temperature in Fahrenheit is interval; number of students in a class is ratio.

Sampling Methods

Types of Sampling

Sampling methods determine how data are collected from a population, affecting the validity of statistical conclusions.

• Cluster Sampling: Divide the population into clusters, randomly select clusters, and use all members from selected clusters.

• Convenience Sampling: Use data that are easy to obtain; may introduce bias.

• Random Sampling: Every member of the population has an equal chance of being selected.

• Stratified Sampling: Divide the population into strata (groups) and randomly sample from each stratum.

• Systematic Sampling: Select every k-th member from a list after a random start.

Example: Surveying every 10th person entering a store is systematic sampling.

Types of Data

Discrete vs. Continuous Data

Data can be classified based on the possible values they can take.

• Discrete Data: Countable values (e.g., number of students, number of cars).

• Continuous Data: Any value within a range (e.g., height, weight, time).

Example: The number of books is discrete; the weight of a book is continuous.

Frequency Distributions

Constructing and Interpreting Frequency Tables

Frequency distributions organize data into classes or intervals, showing how many data points fall into each class.

• Class Boundaries: The values that separate classes without gaps.

• Class Midpoints: The average of the upper and lower class boundaries.

• Class Width: The difference between consecutive lower class boundaries.

Example: For classes 10-19, 20-29, the class midpoint for 10-19 is (10+19)/2 = 14.5.

Relative and Cumulative Frequency Distributions

• Relative Frequency: The proportion of data in each class.

• Cumulative Frequency: The sum of frequencies for all classes up to a certain class.

Example Table:

Class	Frequency	Relative Frequency	Cumulative Frequency
10-19	5	0.10	5
20-29	12	0.24	17
30-39	18	0.36	35
40-49	15	0.30	50

Additional info: Table values are illustrative.

Frequency Histograms

A frequency histogram is a bar graph representing the frequency distribution of a dataset.

• Bars touch each other (no gaps) to indicate continuous data.

• Horizontal axis: class midpoints or boundaries.

• Vertical axis: frequency or relative frequency.

• Label axes and start vertical scale at 0.

Describing Distributions

Shapes of Distributions

The shape of a distribution provides information about the data's symmetry and spread.

• Skewed Left: Tail extends to the left; mean < median.

• Skewed Right: Tail extends to the right; mean > median.

• Uniform: All classes have approximately equal frequencies.

• Bell-Shaped: Symmetrical, with most data near the center (normal distribution).

Measures of Central Tendency and Position

Calculating Mean, Median, Mode, and Midrange

• Mean (Arithmetic Average): Sum of all data values divided by the number of values.

• Median: Middle value when data are ordered.

• Mode: Most frequently occurring value.

• Midrange: Average of the maximum and minimum values.

Formula for Mean (from frequency table):

$\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$

Example: If the sum of (frequency × midpoint) is 200 and total frequency is 20, then $\bar{x} = 200/20 = 10$.

Percentiles

• Percentile: The value below which a given percentage of data falls.

• To find the percentile of a data value, use its position in the ordered data set.

Example: The 75th percentile (P75) is the value below which 75% of the data fall.

Measures of Variation

Range, Variance, and Standard Deviation

• Range: Difference between the maximum and minimum values.

• Variance: Average of the squared differences from the mean.

• Standard Deviation: Square root of the variance; measures spread of data.

Formula for Sample Standard Deviation:

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

Relationship: $s^2$ is the variance.

Z-Scores and Outliers

Calculating and Interpreting Z-Scores

• Z-Score: Number of standard deviations a data value is from the mean.

• Formula: $z = \frac{ x - \bar{x} }{ s }$

• To find a data value from a z-score: $x = \bar{x} + z s$

• Values with z-scores less than -2 or greater than 2 are often considered significantly low or high.

Example: If $\bar{x} = 50$, $s = 10$, and $x = 70$, then $z = (70-50)/10 = 2$.

Probability

Basic Probability Concepts

• Simple Probability: Probability of a single event occurring.

• Formula: $P(A) = \frac{ \text{Number of favorable outcomes} }{ \text{Total number of outcomes} }$

Addition Rule ("OR" Probability)

• For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$

• For non-mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Multiplication Rule ("AND" Probability)

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

• For dependent events, adjust probabilities as needed.

Example: Drawing two cards without replacement requires adjusting the denominator for the second draw.

Probability of "At Least One"

• "At least one" means one or more occurrences.

• Formula: $P(\text{at least one}) = 1 - P(\text{none})$

Example: Probability of getting at least one head in three coin tosses: $1 - (1/8) = 7/8$.

Best Practices for Calculations

• Show the values used in formulas for mean, standard deviation, probability, z-score, and percentiles.

• It is not necessary to show every arithmetic step, but clarity in the process is important.