Levels of Measurement

Types of Measurement Scales

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

• 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, but no true zero point (e.g., temperature in Celsius).

• Ratio: Ordered data with meaningful differences and a true zero point (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:

• 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 cars, number of students).

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

Example: The number of books on a shelf is discrete; the weight of a book 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 is longer; mean < median.

• Skewed Right: Tail on the right side is longer; mean > median.

• Uniform: All classes have approximately equal frequencies.

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

Measures of Central Tendency and Position

Calculating the Mean

• For grouped data, use the formula:

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

• Where f is the frequency and x is the class midpoint.

Other Measures

• Median: Middle value when data are ordered.

• Mode: Most frequently occurring value.

• Midrange: Average of the highest and lowest values.

• Range: Difference between the highest and lowest values.

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

Finding Percentiles

• To find the percentile of a data value, use:

$ \text{Percentile} = \frac{\text{Number of values below}}{\text{Total number of values}} \times 100 $

• To find the data value at a given percentile, use the position formula:

$ L = \frac{k}{100} \times n $

where L is the location, k is the percentile, and n is the number of data values.

Measures of Variation

Variance and Standard Deviation

• Variance (s2): Average squared deviation from the mean.

• Standard Deviation (s): Square root of the variance.

To calculate standard deviation for a sample:

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

• Given the standard deviation, variance is $ s^2 $.

Z-Scores

Standardizing Data

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

Formula:

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

• To find a data value given a z-score:

$ x = z \cdot s + \bar{x} $

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

Probability

Basic Probability

• Simple Probability: Probability of a single event occurring.

$ 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 overlapping outcomes.

Multiplication Rule ("AND" Probability)

• For independent events:

$ P(A \text{ and } B) = P(A) \times P(B) $

• For dependent events, adjust probabilities as needed.

Probability of "At Least One"

• Probability that at least one event occurs:

$ 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.