Levels of Measurement

Types of Measurement Scales

Understanding the level of measurement is essential for selecting appropriate statistical methods. There are four main levels:

• 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; height in centimeters is ratio.

Sampling Methods

Types of Sampling

Sampling methods determine how data are collected from a population:

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

• Stratified Sampling: The population is divided into subgroups (strata) and random samples are taken from each stratum.

• Cluster Sampling: The population is divided into clusters, some clusters are randomly selected, and all members of chosen clusters are sampled.

• Systematic Sampling: Every k-th member of the population is selected after a random starting point.

• Convenience Sampling: Samples are taken from a group that is easy to access, which may introduce bias.

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

Types of Data

Discrete vs. Continuous Data

• Discrete Data: Countable values (e.g., number of students in a class).

• Continuous Data: Measurable values that can take any value within a range (e.g., height, weight).

Example: The number of cars in a parking lot is discrete; the time it takes to run a mile is continuous.

Frequency Distributions

Key Components

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

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

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

Constructing Frequency Tables

• Relative Frequency Distribution: Shows the proportion of data in each class.

• Cumulative Frequency Distribution: Shows the total number of data values less than or equal to the upper class boundary.

Example Table: Relative Frequency Distribution

Frequency Histograms

• Bar graphs representing frequency distributions.

• No gaps between bars for continuous data.

• Horizontal axis can use class midpoints or boundaries.

• Label axes and start vertical scale at 0.

Describing Distributions

Shapes of Distributions

• Skewed Left: Tail on the left side.

• Skewed Right: Tail on the right side.

• Uniform: All classes have approximately equal frequencies.

• Bell-Shaped: Symmetrical, highest frequency in the middle (normal distribution).

Descriptive Statistics

Measures of Central Tendency

• 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 highest and lowest values.

Measures of Variation

• Range: Difference between the highest and lowest values.

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

• Standard Deviation: Square root of the variance.

Formulas

• Mean (for frequency table): $\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$

• Sample Standard Deviation: $s = \sqrt{\frac{n(\sum x^2) - (\sum x)^2}{n(n-1)}}$

Percentiles

• Percentiles indicate the relative standing of a value within a data set.

• To find the percentile of a value: $\text{Percentile} = \frac{\text{Number of values below}}{\text{Total number of values}} \times 100$

• To find the data value for a given percentile, use the formula: $L = \frac{k}{100} \times n$ (where $L$ is the location, $k$ is the percentile, $n$ is the number of data values).

Z-Scores

Definition and Calculation

• Z-Score: Measures how many standard deviations a data value is from the mean.

• Formula: $z = \frac{x - \mu}{\sigma}$

• For sample data, use $\bar{x}$ and $s$ instead of $\mu$ and $\sigma$.

• To find a data value from a z-score: $x = z \cdot \sigma + \mu$

• Values with $|z| > 2$ are often considered significantly low or high.

Probability

Basic Probability

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

Addition Rule ("OR" Probability)

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

• Do not double count outcomes that are in both A and B.

Multiplication Rule ("AND" Probability)

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

• For dependent events, adjust the probability for the second event.

Probability of "At Least One"

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

• Use the complement rule: $P(\text{at least one}) = 1 - P(\text{none})$

Best Practices for Calculations

• Show your work for mean, standard deviation, probability, z-score, and percentiles.

• Include the values used in formulas, but not every arithmetic step.