Chapter 2- frequency distributions PDF

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Frequency distributions 2.1 graphically summarizing quantitative data ● A (relative) frequency histogram for a quantitative data set is a bar graph in which the height of the bar shows “how often” (measured as a proportion or relative frequency) measurements fall in a particular class or subinterval Frequency distributions and histograms ● Frequency distribution ○ A summary of a set of data that displays the number of observations in each of the distribution’s distinct categories or classes ○ Is a list or a table ○ Contains the values of a variable (or a set of ranges within which the data fall) ○ Also contains the corresponding frequencies with each value occurs (or frequencies with which data fall within each range) Discrete data (ungrouped) ● Data that can take on a countable number of possible values ○ An advertiser asks 2000 customers how many days per week they read the daily newspaper Relative frequency ● The proportion of total observations that are in a given category

Developing frequency distribution for distribution for discrete data ● Step 1: lost the possible values ● Step 2: count the number of occurrences at each value

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Step 3: determine the relative frequencies

Proportions and percentages ● Proportions ○ Measures the fraction of the total group that is associated with each score ○ Proportion p=f/n ○ Called relative frequencies because they describe the frequency (f) in relations to the total number (N) ● Percentages ○ Expresses relative frequency out of 100 ○ Percentage p(100)=f/n(100) ○ Can be included as a separate column in a frequency distribution table Learning check 1 (slide 1 of 4) ● Use the frequency distribution table to determine how many subjects were in the study ○ 10 ○ 15 ○ 33 ○ Impossible to determine

Use picture for both question **

Learning Check 1 (slide 3 of 4) ● For the frequency distribution shown, is statements true or false?

each of these

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More than 50% of the individuals score above 3 (True) The proportion of scores in the lowest category was p=3 (false)

Grouped frequency distribution tables ● If the number of categories is very large, they are combined (grouped) to make the table easier to understand ● However, information is lost when categories are grouped ○ Individual scores cannot be retrieved ○ The wider the grouping interval, the more information is lost Group data ● Continuous data ○ Data whose possible values are uncountable and that may assume any value in an interval (weight, length, time) ○ Discrete data with many possible outcomes (age, income, stock price) ○ Summarized in a grouped data frequency distribution ○ Data are organized in classes “Rules” for constructing grouped frequency distributions ● Requirements (mandatory guidelines) ○ All intervals must be the same width ○ Make the bottom (low) score in each interval a multiple of the interval width ● “Rules of thumb” (suggested guidelines) ○ No. of classes: ten or fewer class intervals is typical (but use good judgement for the specific situation) (alternative: Sturges rule: 1 + 3.322*log10 (N) and finally round off to a convenient number) (use the ‘log’ button (not ‘in’ in your calculator) ○ Class width: choose a “simple” number for interval width (e.g, 2,5,10) (alternative: (max-min)/ No. of classes and finally round up) ● Step 1: no. of classes (10 or fewer/sturge’s rule) ● Step 2: class width (convenient number or (Max-Min)/No. Of classes and then round) ● Step 3: construct classes using apparent class limits and ensure that the bottom (low) score in each interval a multiple of the interval width ● Step 4: set real limits (subtract 0.5 from lower apparent lower limit and add 0.5 to apparent upper limit) ● Step 5: tally, frequency, and relative frequency Example 2.5: grouped frequency distribution ● An instructor has obtained the set of N -- 25 exam scores shown here. To help organize these scores, we will place them in a frequency distribution table. The scores are:

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Step 1: No. of classes = 9 (alternative: sturges rule: 1+3.322 * log10(25)= 5.64 =6 or

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above Step 2: class width = 5 (alternative: (94-53)/9=4.55=5) Step 3: apparent class limits:50-54, 55- 59, …, 90-94 Step 4: real limits: 49.5-54.5, 54.5-59.5, …, 89.5-94.5

Learning check 2 (slide 1 of 4) ● A grouped frequency distribution table has categories 0-9,10-19, 20-29, and 30-39. What is the width of the interval 20-29 ○ 9 points ○ 9.5 points ○ 10 points (29.5-19.5 = 10) ○ 10.5 points ● Decide if each of the following statements is true or false ○ You can determine how many individuals had each score from a frequency distribution table (true) ○ You can determine how many individuals had each score from a grouped frequency distribution (false) 2.3 frequency distribution graphs ● Pictures of the data organized in tables ○ All have two axes ○ X-axis (abscissa) typically has categories of measurement scale increasing left to right ○ Y-axis (ordinate) typically has frequencies increasing bottom to top ● General principles

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Both axes should have value 0 where they meet Height should be about ⅔ to ¾ of length

Graph the histogram ● Rectangles represent the classes ● The base represents the class length ● The height represents ○ The frequency in a frequency histogram ○ The relative frequency in a relative frequency histogram Data graphing questions ● Level of measurement (nominal;ordinal; interval;or ratio) ● Discrete or continuous data) ● Describing samples or populations? ○ The answers to these questions determine which is the appropriate graph Frequency distribution histogram (quantitative variables-- interval or ratio level data) (slide 1 of 2) ● Requires numeric scores (interval or ratio) ● Represent all scores on X-axis from minimum thru maximum observed values ● Include all scores with frequency of zero ● Draw bars above each score (interval) ○ Height of bar corresponds to frequency ○ Width of bar corresponds to score real limits (or one-half score unit above/below discrete scores) Figure 2.1 (ungrouped) frequency distribution histogram

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Same requirements as for frequency distribution histogram expect: Draw bars above each (grouped) class interval ○ Bar width is the class interval real limits ○ Consequences? Apparent limits are extended out one-half score unit at each end of the interval

Figure 2.2: grouped frequency distribution histogram

Box 2.1 - figure 2.4 use and misuse of graphs

Frequency distribution polygons ● List all numeric scores on the X-axis ○ Include those with a frequency of f=0 ● Draw a dot above the center of each interval ○ Height of dot corresponds to frequency ○ Connects the dots with a continuous line ○ Close the polygon with lines to the Y=o points ● Can also be used with grouped frequency distribution data Figure 2.5: frequency distribution polygon (ungrouped data)

Figure 2.6: grouped data frequency distribution polygon

Graphing summarizing qualitative data ● With qualitative data, names identify the different categories ● This data can be summarized using a frequency distribution ● Frequency distribution: a table that summarizes the number of items in each of several non overlapping classes Examples 2.1: describing pizza preferences ● Table 2.1 lists pizza preferences of 50 college students

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Table 2.2 does not reveal much useful information A frequency distribution is a useful summary

Relative frequency and present frequency ● Relative frequency summarizes the proportion of items in each class ● For each class, divide the frequency of the class by the total number of observations ● Multiply times 100 to obtain the percent frequency

Graphically summarizing qualitative data ● Use a data distribution to describe: ○ What values of the variable have been measured ○ How often each value has occurred ● “How often” can be measured in 3 ways: ○ Frequency ○ Relative frequency= frequency/n (n=sample size) ○ Percent = 100x relative frequency Bar charts ● A graphical representation of a categorical data set in

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which a rectangle or bar is drawn over each category or class The length or height of each bar represents the frequency or percentage of observations or some other measure associated with the category The bars may be vertical or horizontal Bar charts are normally drawn in vertical columns, although they can also be drawn with horizontal bars

Constructing bar chart ● Step 1: define the categories for the variable of interest ● Step 2: for each category, determine the appropriate measure or value ● Step 3: for a column bar chart, locate the categories on the horizontal axis. For a horizontal bar chart, place the categories on the vertical axis. Then construct bars, either vertical or horizontal, for each category such that the length or height corresponds to the value for the category ● Step 4: interpret the results The area principle ● Although the height of each column corresponds to the correct relatives frequency of loblaw stores, our eyes tend to be impressed by the area, which can be misleading

The area principle ● The best displays observe the area principle: the area occupied by a part of the graph should correspond to the magnitude of the value it represents Pie charts ● A graph in the shape of a circle ● The circle is divided into “slices” corresponding to the categories or classed to be displayed ● The size of each slice is proportional to the magnitude of the displayed variable associated with each category or class

Pie chart example

Constructing pie chart ● Step 1: define the categories for the variable of interest ● Step 2: for each category, determine the appropriate measure or value. The value assigned to each category is the proportion the category is to the total for all categories ● Step 3: construct the pie chart by displaying one slice for each category that is proportional in size to the proportion the category value is to the total of all categories. Sector angle = (f/n)*306 degrees Figure 2.10: distribution shapes

Interpreting graphs: outliers

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Are there any strange or unusual measurements that stand out in the data set?

Learning check 3 (slide 1 of 4) ● What is the shape of this distribution ○ Symmetrical ○ Negatively skewed ○ Positively skewed ○ Discrete ● Decide if each of the following statements is true or false ○ It would be correct to use a histogram to graph parental marital status data (single, married, divorced) from a treatment center for children (false) ○ It would be correct to use a histogram to graph the time children spent playing with other children from data collected in a children's treatment center (true)...