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2007© BOLD Educational Software Descriptive Statistics: Frequency Distributions and Graphic Presentations Note: for more information, print the NOTES PAGES, or view the notes below the PowerPoint slide

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2007© BOLD Educational Software What Is the Question? Descriptive statistics answers the question: What does this data look like? Descriptive statistics answers the question: What does this data look like?

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2007© BOLD Educational Software Frequency Distribution Frequency distribution: a list of all the scores for a sample or population indicating the number of times that each score occurs. The data are grouped into mutually exclusive categories showing the number of observations in each category. Frequency distribution: a list of all the scores for a sample or population indicating the number of times that each score occurs. The data are grouped into mutually exclusive categories showing the number of observations in each category.

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2007© BOLD Educational Software Practice: Describe the Distribution 46 students took an 18-question quiz. Describe the distribution. Note: read zero, up to, but not including 3; 3 up to, but not including 6, etc. 46 students took an 18-question quiz. Describe the distribution. Note: read zero, up to, but not including 3; 3 up to, but not including 6, etc. Studentsf 0 up to 31 3 up to 61 6 up to 94 9 up to up to up to 1825 Total46

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2007© BOLD Educational Software Practice: Describe the Distribution Out of 46 students, largest number of students (25) got 15, 16, 17, or 18 answers correct. Only 2 students got less than 6 answers correct. Studentsf 0 up to 31 3 up to 61 6 up to 94 9 up to up to up to 1825 Total46

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2007© BOLD Educational Software Creating a Relative Frequency Distribution StudentsfRelative Frequency 0 up to %1/46 3 up to /46 6 up to /46 9 up to /46 12 up to /46 15 up to /46 Total

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2007© BOLD Educational Software Practice: Where Do the Data Tend to Cluster? Data tend to cluster between 40 and 59 Agesf 10 up to up to up to up to up to up to up to 804 Total60

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2007© BOLD Educational Software Practice: Describe the Distribution The ages range from a low of 10 to a high of 79, with some clustering in the 40 to 59 age brackets. Agesf 10 up to up to up to up to up to up to up to 794 Total60

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2007© BOLD Educational Software AgesfRelative Freq. 10 up to 192 2/60 20 up to 291 1/60 30 up to up to up to up to up to 794 Total Practice: Determine the Relative Frequency Distribution

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2007© BOLD Educational Software Charts and Graphs

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2007© BOLD Educational Software Graphic Presentation of a Frequency Distribution The three commonly used graphic forms are: –Histograms, – Bar charts, –Frequency polygons. The three commonly used graphic forms are: –Histograms, – Bar charts, –Frequency polygons.

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2007© BOLD Educational Software Graphic Presentation of a Frequency Distribution Histogram: A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.

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2007© BOLD Educational Software Histogram for Years of Experience: Note the Normal Curve Superimposed

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2007© BOLD Educational Software Symmetric When folded vertically, both sides are (more or less) the same. Common Shapes of Histograms

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2007© BOLD Educational Software Also Symmetric Common Shapes of Histograms

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2007© BOLD Educational Software Uniform (Platykurtic – flat) Common Shapes of Histograms

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2007© BOLD Educational Software Leptokurtic (high peak)

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2007© BOLD Educational Software Non-Symmetric Histograms. These histograms are skewed. Common Shapes of Histograms

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2007© BOLD Educational Software Skewed Histograms Skewed left (negative skew) Skewed right (positive skew) Common Shapes of Histograms

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2007© BOLD Educational Software Skewed Histograms Skewed left (negative skew) Skewed right (positive skew) Notice that the SKEW follows the TAIL

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2007© BOLD Educational Software Common Shapes of Histograms Bimodal f The two largest rectangles are separated by at least one class.

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2007© BOLD Educational Software Bar Graphs A bar graph illustrates nominal data with the scores/categories along the x-axis and the frequencies on the y-axis. The scores are not ordered. The bars do not touch (unlike a histogram). The heights correspond to the number of times the score occurs. A bar graph illustrates nominal data with the scores/categories along the x-axis and the frequencies on the y-axis. The scores are not ordered. The bars do not touch (unlike a histogram). The heights correspond to the number of times the score occurs.

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2007© BOLD Educational Software Practice: Describe the Data The majority of students in the sample were White. The next largest group was Hispanic. The smallest representative groups were Black, Asian, and Indian.

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2007© BOLD Educational Software Graphic Presentation of a Frequency Distribution A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.

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2007© BOLD Educational Software Frequency Polygon for Hours Spent Studying Hours spent studying Frequency

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2007© BOLD Educational Software Compare the Frequency Polygon to the Histogram

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2007© BOLD Educational Software Histogram Using SPSS

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2007© BOLD Educational Software Frequency Polygon Using SPSS

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2007© BOLD Educational Software Stem and Leaf A frequency distribution table that provides a visual picture of the distribution

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2007© BOLD Educational Software Stem and Leaf Each raw score has two parts: a stem, consisting of all but the last digit, and the leaf, the last digit in the number.

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2007© BOLD Educational Software Stem and Leaf Current Salary Stem-and-Leaf Plot Frequency Stem & Leaf Extremes (>=81250) Stem width: Each leaf: 1 case(s)

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2007© BOLD Educational Software Current Salary Stem-and-Leaf Plot Frequency Stem & Leaf Extremes (>=81250) Stem width: Each leaf: 1 case(s) Each stem represents 10 thousand, so the 1 (stem) = 10,000 There are two cases (frequency=2) with 15,000, two cases with 20,000 (actually, 24,000 and 27,000), 6 cases with 30,000 (30, 30, 31, 32, 33, and 34 thousand) in this data set. Each stem represents 10 thousand, so the 1 (stem) = 10,000 There are two cases (frequency=2) with 15,000, two cases with 20,000 (actually, 24,000 and 27,000), 6 cases with 30,000 (30, 30, 31, 32, 33, and 34 thousand) in this data set.

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2007© BOLD Educational Software Stem and Leaf in SPSS To create a stem and leaf in SPSS, select the following: Analyze Descriptives Explore Select stem and leaf in plots Click continue Click OK To create a stem and leaf in SPSS, select the following: Analyze Descriptives Explore Select stem and leaf in plots Click continue Click OK

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2007© BOLD Educational Software Scatter plots A scatter plot illustrates the relationship between two continuous variables.

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2007© BOLD Educational Software A scatter plot illustrates the values of Y (vertical axis) versus the corresponding values of X (horizontal axis) Scatter plots

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2007© BOLD Educational Software Scatter plots can provide answers to the following questions: Are variables X and Y correlated? (as one variable goes up, the other variable goes up/down) Scatter plots can provide answers to the following questions: Are variables X and Y correlated? (as one variable goes up, the other variable goes up/down) Scatter plots

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2007© BOLD Educational Software Scatter plots can provide answers to the following questions: Is there a linear relationship between X and Y? (as one variable goes up, the other variable goes up/down) Scatter plots can provide answers to the following questions: Is there a linear relationship between X and Y? (as one variable goes up, the other variable goes up/down) Scatter plots

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2007© BOLD Educational Software Scatter plots Scatter plots can provide answers to the following questions: Is there a curvilinear relationship between variables X and Y? (As Y goes up X goes up, then at a peak, as X continues to go up, Y goes down Scatter plots can provide answers to the following questions: Is there a curvilinear relationship between variables X and Y? (As Y goes up X goes up, then at a peak, as X continues to go up, Y goes down

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2007© BOLD Educational Software Scatter plots Scatter plots can provide answers to the following questions: Are there outliers? (Do one or more points stray from the trend?) Scatter plots can provide answers to the following questions: Are there outliers? (Do one or more points stray from the trend?)

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