Frequency Distributions and Percentiles

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Presentation transcript:

Frequency Distributions and Percentiles Chapter Three Frequency Distributions and Percentiles

New Statistical Notation The number of times a score occurs is the score’s frequency, which is symbolized by f A distribution is the general name for any organized set of data N is the sample size indicating the number of scores Copyright © Houghton Mifflin Company. All rights reserved.

Simple Frequency Distributions Copyright © Houghton Mifflin Company. All rights reserved.

Simple Frequency Distribution A simple frequency distribution shows the number of times each score occurs in a set of data The symbol for a score’s simple frequency is simply f Copyright © Houghton Mifflin Company. All rights reserved.

Raw Scores Following is a data set of raw scores. We will use these raw scores to construct a simple frequency distribution table. 14 13 15 11 10 12 17 Copyright © Houghton Mifflin Company. All rights reserved.

Simple Frequency Distribution Table Copyright © Houghton Mifflin Company. All rights reserved.

Graphing a Simple Frequency Distribution A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use A bar graph A histogram A polygon Copyright © Houghton Mifflin Company. All rights reserved.

A Simple Frequency Bar Graph Is Used for Nominal and Ordinal Data. Copyright © Houghton Mifflin Company. All rights reserved.

A Histogram Is Used for a Small Range of Different Interval or Ratio Scores. Copyright © Houghton Mifflin Company. All rights reserved.

A Frequency Polygon Is Used for a Large Range of Different Scores Copyright © Houghton Mifflin Company. All rights reserved.

Types of Simple Frequency Distributions Copyright © Houghton Mifflin Company. All rights reserved.

The Normal Distribution A bell-shaped curve Called the normal curve or a normal distribution It is symmetrical The far left and right portions containing the low-frequency extreme scores are called the tails of the distribution Copyright © Houghton Mifflin Company. All rights reserved.

An Ideal Normal Distribution Copyright © Houghton Mifflin Company. All rights reserved.

A distribution may be either negatively skewed or positively skewed Skewed Distributions A skewed distribution is not symmetrical as it has only one pronounced tail A distribution may be either negatively skewed or positively skewed Whether a skewed distribution is negative or positive corresponds to whether the distinct tail slopes toward or away from zero Copyright © Houghton Mifflin Company. All rights reserved.

Negatively Skewed Distribution A negatively skewed distribution contains extreme low scores that have a low frequency, but does not contain low frequency extreme high scores Copyright © Houghton Mifflin Company. All rights reserved.

Positively Skewed Distribution A positively skewed distribution contains extreme high scores that have a low frequency, but does not contain low frequency extreme low scores Copyright © Houghton Mifflin Company. All rights reserved.

Bimodal Distribution A bimodal distribution is a symmetrical distribution containing two distinct humps Copyright © Houghton Mifflin Company. All rights reserved.

Rectangular Distribution A rectangular distribution is a symmetrical distribution shaped like a rectangle Copyright © Houghton Mifflin Company. All rights reserved.

Relative Frequency and the Normal Curve Copyright © Houghton Mifflin Company. All rights reserved.

Relative frequency is the proportion of time the score occurs The symbol for relative frequency is rel. f The formula for a score’s relative frequency is rel. f = Copyright © Houghton Mifflin Company. All rights reserved.

A Relative Frequency Distribution Copyright © Houghton Mifflin Company. All rights reserved.

Finding Relative Frequency Using the Normal Curve The proportion of the total area under the normal curve that is occupied by a group of scores corresponds to the combined relative frequency of those scores. Copyright © Houghton Mifflin Company. All rights reserved.

Computing Cumulative Frequency and Percentile Copyright © Houghton Mifflin Company. All rights reserved.

Cumulative Frequency Cumulative frequency is the frequency of all scores at or below a particular score The symbol for cumulative frequency is cf To compute a score’s cumulative frequency, we add the simple frequencies for all scores below the score with the frequency for the score Copyright © Houghton Mifflin Company. All rights reserved.

A Cumulative Frequency Distribution Copyright © Houghton Mifflin Company. All rights reserved.

Percentile A percentile is the percent of all scores in the data that are at or below the score If the scores cumulative frequency is known, the formula for finding the percentile is Score’s Percentile = Copyright © Houghton Mifflin Company. All rights reserved.

Finding Percentiles The percentile for a given score corresponds to the percent of the total area under the curve that is to the left of the score. Copyright © Houghton Mifflin Company. All rights reserved.

Percentiles Normal distribution showing the area under the curve to the left of selected scores. Copyright © Houghton Mifflin Company. All rights reserved.

Grouped Frequency Distributions Copyright © Houghton Mifflin Company. All rights reserved.

Grouped Distributions In a grouped distribution, scores are combined to form small groups, and then we report the total f, rel. f, or cf of each group Copyright © Houghton Mifflin Company. All rights reserved.

A Grouped Distribution Copyright © Houghton Mifflin Company. All rights reserved.

Example 1 Using the following data set, find the relative frequency of the score 12 14 13 15 11 10 12 17 Copyright © Houghton Mifflin Company. All rights reserved.

The simple frequency table for this set of data is as follows. Example 1 The simple frequency table for this set of data is as follows. Copyright © Houghton Mifflin Company. All rights reserved.

The frequency for the score of 12 is 1. N = 18. Example 1 The frequency for the score of 12 is 1. N = 18. Therefore, the rel. f of 12 is Copyright © Houghton Mifflin Company. All rights reserved.

Example 2 What is the cumulative frequency for the score of 14? Answer: The cumulative frequency of 14 is the frequency of all scores at or below 14 in this data set cf = 1 + 1 + 1 + 4 + 6 = 13 The cf for the score of 14 in this data set is 13 Copyright © Houghton Mifflin Company. All rights reserved.

Example 3 What is the percentile for the score of 14? Answer: The percentile of 14 is the percentage of all scores at or below 14 in this data set 0.056 + 0.056 + 0.056 + 0.222 + 0.333 = 0.72 Another way to calculate this percentile is Copyright © Houghton Mifflin Company. All rights reserved.