Basic Statistics for the Behavioral Sciences - Chapter Three

Basic Statistics for the Behavioral Sciences - Chapter Three: Frequency Distributions and Percentiles

Introduction

• Before analyzing the relationship between two variables, it is essential to summarize each variable independently.

• Key Questions:

  • Which scores occurred in the data?

  • How often did each score occur?

• This information is organized into tables and graphs using frequency distributions.

New Terms to Know

• Frequency (f): The number of occurrences of a particular score in a dataset.

• Distribution: A general term for any organized set of data.

• Sample Size (N): Indicates the total number of scores in the dataset.

Simple Frequency Distributions

• A simple frequency distribution displays the frequency of each score in a dataset.

  • Example: Count the number of times “Male” appears in class responses.

• The symbol used for a score’s simple frequency is f.

Constructing a Simple Frequency Distribution Table

• Typically involves:

  • Listing the highest and lowest scores.

  • Including all scores, with zeros for unpicked scores.

  • Important Note:

  • N is not simply the sum of scores but the count of individual data points (sum of frequencies, f).

Graphing a Simple Frequency Distribution

• A frequency distribution graph presents scores on the X-axis and their frequencies on the Y-axis.

• The type of measurement scale (nominal, ordinal, interval, or ratio) influences the graph type to use:

  • Bar Graph: Used for nominal and ordinal data (discrete categories, bars do not touch).

  • Histogram: Used for small ranges of interval or ratio scores (continuous data, bars touch).

  • Frequency Polygon: Used when dealing with large ranges of scores (points connected by lines).

Distribution Types

• Various types of frequency distributions identified:

  • Normal Distribution:

  • Identified by a bell-shaped curve.

  • Symmetrical, with tails on either side containing low-frequency scores.

  • Skewed Distributions:

  • Negatively Skewed: More frequent low scores; the tail points leftward.

  • Positively Skewed: More frequent high scores; the tail points rightward.

  • Bimodal Distribution: A symmetrical distribution with two distinct frequency peaks (humps).

  • Rectangular Distribution: A symmetrical shape without tails, featuring uniform frequency across all values.

Relative Frequency and Normal Curve

• Relative Frequency (rel.f): The proportion of occurrences of a score (odds in terms of the total).

• The formula for calculating relative frequency:

  • $\text{rel. f} = \frac{f}{N}$

  • Where:

  • f = frequency of the score

  • N = total number of scores

• The area under the normal curve for a group of scores correlates to their combined relative frequency.

Cumulative Frequency and Percentiles

• Cumulative Frequency (cf): The total frequency of all scores at or below a particular score.

• To compute cumulative frequency:

  • Sum all frequencies for scores at and below the target score.

• Percentile: The percentage of all scores that are at or below a given score.

  • For instance, being in the 90th percentile means scoring better than 90% of participants.

• The formula to find a score's percentile is:

  • $\text{Score's Percentile} = \frac{cf}{N} \times 100$

Grouped Frequency Distributions

• When dealing with extensive datasets, it might be necessary to combine scores into small groups, thus creating a grouped frequency distribution.

• This technique reports total frequencies, relative frequencies, or cumulative frequencies for each group, saving space.

Examples and SPSS Applications

• Using datasets to find relative frequencies, cumulative frequencies, and percentiles are explained through examples.

• SPSS: A statistical software program that can perform detailed analyses on datasets, similar to Excel but focused on statistics.

  • Scenario example: Analyzing college students' perceptions of friendship ease with gender considerations.

  • Analyzing family incomes of incoming freshmen to determine income distributions.

• Example outputs can include:

  • Total sample size answering specific questions, percentages encountering gaps in data, and implications for findings based on demographic responses.