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Chapter 2 Notes 2.1 Organizing and Graphing Qualitative Data 2.1.1 Raw Data ⋅ when data is collected, the information collected from each member is recorded in the sequence in which it becomes available • This sequence of data recording is random and unranked

raw data 一 data recorded in the sequence in which they are collected and before they are processed or ranked ungrouped data 一 a data set that contains information on each member individually 2.1.2 Frequency Distributions frequency distributions of a qualitative variable 一 lists all categories and the number of elements that belong to each category How to Construct a Frequency Distribution Table: 1.

find all the categories for the variable and record them in the first column

2.

read each response and mark a tally in the second column next to the corresponding category

3.

record the total number of tallies for each category in the third column

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this column is called the column of frequencies and is denoted by f.

⋅ the sum of the entries in the frequency column gives the sample size or total frequency 2.1.3 Relative Frequency and Percentage Distributions ⋅ the relative frequency of a category is obtained by dividing the frequency of that category by the sum of all frequencies • a relative frequency distribution lists the relative frequency for all categories

relative frequency of a category = frequency of that category sum of all frequency percentage = (relative frequency) • 100% How to Determine the Relative Frequency and Percentage Distributions:

1.

divide the frequency from each category by the sum of all frequencies 一 the relative frequency

4.

multiply the relative frequency by 100 一 percentage

⋅ the sum of all relative frequencies is always 1 and the sum of all percentages is always 100 2.1.4 Graphical Presentation of Qualitative Data Bar Graphs ⋅ various categories are marked on the horizontal axis • each category is represented by intervals of the same width ⋅ each frequency is marked on the vertical axis ⋅ draw one bar for each category where the height represents the frequency • leave a small gap between adjacent bars

bar graphs 一 a graph made of bars whose heights represent the frequencies of respective categories ⋅ bar graphs for relative frequency and percentage distribution can be drawn by marking the relative frequencies or percentages, instead of the frequencies on the vertical axis ⋅ sometimes a bar graph can be constructed by making the categories on the vertical axis and the frequencies on the horizontal axis Pareto Chart ⋅ the whole circle represents the total sample or population ⋅ it is divided into different portions to represent different categories pie chart 一 a circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories

2.2 Organizing and graphing Quantitative Data

2.2.1 Frequency Distributions ⋅ an interval that includes all values that fall on or within two numbers - the upper and lower limits - these classes always represent a variable ⋅ classes are non-overlapping - each value belongs to only one class ⋅ each class has a lower limit (small number) and an upper limit (higher number)

frequency distribution for quantitative data 一 lists all classes and the number of values that belong to each class • data from this is called grouped data

⋅ the difference between the lower limits of two consecutive classes gives the class width (aka. class size ) How to Calculate Class Width: class width = lower limit of next class - lower limit of current class ⋅ the class midpoint (aka. mark) is obtained by dividing the sum of the two limits of a class by 2 How to Calculate Class Midpoint: class midpoint = lower limit + upper limit 2 2.2.2 Constructing Frequency Distribution Tables 1.

find the number of classes - usually vary from 5-20

5.

It is preferred for class widths to all be the same size. After the calculation round it to a convenient number

approximate class width = largest value - smallest value number of classes 6.

find the lower limit of the first class - any number that is lower than or equal to the smallest number in the data set

⋅ an equation to help decide on the number of classes is: c = 1 + 3.3logn c = number of classes

n = number of observations

How to Construct a Frequency Distributions Table: 1.

determine how many classes

7.

determine class width

8.

record all classes in the first column

9.

tally the data for the corresponding class in second column

10. count tallies and put it in the third column (frequencies)

⋅ using the ∑ notation to denote the sum of all frequencies for all classes by ∑f 2.2.3 Graphing Grouped Data Histogram ⋅ can be drawn for a frequency distribution, a relative frequency distribution, or a percentage distribution ⋅ it can be called a frequency histogram, a relative frequency histogram, or a percentage histogram depending what is marked on the vertical axis histogram 一 a bar graph which classes are marked on the horizontal axis and the frequencies, relative frequencies or percentages are marked on the vertical axis How to Construct a Histogram: 1.

mark classes on the horizontal axis

11. mark frequencies, relative frequencies or percentages on the vertical axis 12. draw a bar for each class representing the frequency •

Bars are adjacent with no gaps

Polygons ⋅ a polygon is called a frequency polygon, relative frequency polygon or percentage polygon depending on the vertical axis ⋅ used as another way to describe quantitative data How to Construct a Polygon: 1. mark classes on the horizontal axis 13. mark frequencies on the vertical axis 14. mark a dot above the midpoint of each class at a height equal to the frequency of each class 15. include two or more classes, one at each end, and mark their midpoints at zero 16. join the adjacent dots with straight lines

polygon 一 a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines ⋅ for a very large data set, as the number of classes is increased (and the width of classes is decreased), the frequency polygon eventually becomes a smooth curve • this curve is called a frequency distribution curve or a frequency curve

2.2.5 More on Classes and Frequency Distributions Less-Than Method for Writing Classes ⋅ the less-than method is more appropriate when a data set contains fractional values ⋅ instead of limits, the less than method uses boundaries • lower limit = lower boundary •

upper limit = upper boundary

⋅ when writing the class, we write as lower boundary to less than upper boundary Single-Valued Classes ⋅ use if the observations in a data set assume only a few distinct values 2.2.6 Cumulative Frequency Distributions ⋅ each class in a cumulative frequency distribution table gives the total number of values that fall below a certain value ⋅ quantitative data only ⋅ each class has the same lower limit but a different upper limit cumulative relative frequency 一 gives the total number of values that fall below the upper boundary of each class

cumulative relative frequency = cumulative frequency of a class total observations in the data set cumulative percentage = (cumulative relative frequency) • 100 2.2.7 Shapes of Histograms symmetric 一 identical on both sides of its central point skewed 一 not symmetric, the tail on one side is longer than the tail on the other side • skewed to the right 一 longer tail on the right side •

skewed to the left 一 longer tail on the left side

uniform 一 same frequency for each class

2.2.8 Truncating Axes ⋅ graphs can be used to manipulate data: • changing the scale either on one or both axes •

truncating the frequency axis by starting the frequency at a greater number than zero

2.3 Stem and Leaf Displays

⋅ used to present quantitative data ⋅ by using this display none of the information on individual observations is lost stem and leaf display 一 each value is divided into two portions - a stem and a leaf; the leaves for each stem are shown separately in a display How to Construct a Stem and Leaf Display: 1. split all data into two parts - the steam and the leaf 17. draw a vertical line 18. take the first digit and put it on the left - stem 19. take the second digit and put it on the right - leaf

⋅ the stems are on the left are written in order from least to greatest ⋅ if there are too many stems, we combine the rows • the stems are written as intervals and the leaves are separated by asterisks for each individual stem ⋅ if there are a smaller amount of stems and more leaves we can create split stems • to do this we can split each stem into 2 or 5 parts •

If its split into 2 parts any leaves with a value of 0-4 are placed in the first split stem, while the leaves 5-9 are placed in the second split stem

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If it’s split into 5 parts, leaves with values of 0 to 1 are placed in the first stem, 2 and 3 in the second and so on

2.4 Dotplots

⋅ dotplots can help detect outliers in a data set outliers 一 values that are very small or very large, relative to the majority of the values in a data set How to Create a Dotplot: 1. create a number line starting with the minimum and ending with the maximum values in the data set 20. draw a dot above each value on the number line corresponding to the given data...