Sexual Activity and the Lifespan of Male Fruitflies

1. Sexual Activity and the Lifespan of Male Fruitflies Exploring, Organizing, and Describing, Quantitative Data Essentials Data set characteristics Frequency Distributions Frequency Tables for Quantitative Data Charts & Graphs for Quantitative Data Problem Data Sets for Discrete and Continuous Data

2. Essentials:Quantitative DataKnow this stuff - (stuff: a useful filler term in stats.) • Characteristics of quantitative variables. • Building a quantitative frequency table. • From within a quantitative frequency table, be able to identify: classes, class widths, class midpoints, class limits, boundaries (cutpoints) • Identify and construct appropriate charts/graphs for quantitative data.

3. EXPLORING, ORGANIZING, DESCRIBING, AND COMPARING DATA Before beginning to analyze data, it is important to know three things: 1. Did the data come from a sample or a population? 2. Is the data qualitative or quantitative? 3. In what measurement scale is the data reported?

4. Important Characteristics of a Data Set Center – an “average” value that indicates where the middle of the data is located. Variation – a measure of the amount that the values vary among themselves. Distribution – the “shape” of the distribution of data. Outliers – values that are far away from the majority of values. Time – changing characteristics of data over time.

5. FREQUENCY DISTRIBUTIONS A Frequency Distribution represents the range over which a variable’s values occur. A Frequency Table lists classes (or categories) of values, along with the frequencies (counts) of the number of values that fall into each class. In addition, a frequency table may show cumulativefrequencies, relative frequencies, and cumulative relative frequencies. Frequency Tables are derived from RAW DATA and a TALLY process.

6. Quantitative Frequency Table Terms Class – a grouping of data values Lower Class Limits – the smallest number belonging to a class. Upper Class Limits – the largest number belonging to a class. Class Boundaries – numbers used to separate classes without the gaps created by class limits. (Also referred to as Cutpoints) Class Midpoints – the midpoints of the classes. Class Width – the difference between two consecutive lower class limits or lower class boundaries.

7. Quantitative Frequency Distributions Grouped Frequency Distributions include a series of consecutive values into a Class (grouping) Discrete or Continuous data may be presented Ungrouped (or Single-Value) Frequency Distributions contain a Class (grouping) for each value of the variable Generally, a small number of Discrete values are presented

8. Ungrouped or Single-Value Data ExampleData for: Number of School-Age Children

9. Single-Value Grouped Data Table Each value is represented as a class.

10. Frequency Table of a Quantitative Variable(Grouped Data Example) Old Faithful (length of time in minutes, between eruptions for 200 observations) Classes represent ranges of discrete or continuous values. Here the class values represent the Lower and Upper Class Limits.

11. Histograms A way to graphically represent quantitative discrete and continuous data Horizontal scale represents classes. Vertical scale represents frequencies or relative frequencies (or percents). Heights of the bars correspond to the frequencies (or relative frequencies). Bars are adjacent to each other. That is, there are no gaps between bars, (as occurs with bar charts). • See the Anatomy of Statistics reference sheets for additional examples and discussion.

12. Histogram for Single Value Data Note that each discrete value is represented by a bar equaling the value’s frequency or relative frequency and that the bars touch. Here the class values are the midpoints of the bars.

13. Histogram of Old Faithful Data (Continuous Data) Time between Eruptions of Old Faithful Geyser Here the midpoints of the classes are presented.

14. Anatomy of a Histogram Title Note that there are no spaces between bars. (continuous data) Number of observations. Height of each bar represents the frequency in each class. Number of occurrences (frequencies) are shown on the vertical axis. Empty Class: No data were recorded between 75 and 80. The numbers shown on the horizontal axis are the boundaries of each class. (Also known as cutpoints.) Each bar represents a class. The number of classes is usually between 5 and 20. Here, there are 17 classes. The width of each class is determined by dividing the range of the data set by the number of classes, and rounding up. In this data set, the range is 82. 82/17 = 4.8, rounded up to 5. This class goes from 5 to 10. Label both horizontal and vertical axes. NOTE: Sometimes the numbers shown on the horizontal axis are the midpoints of each class. (A class midpoint is also referred to as the mark of the class.)

15. Dotplot • Distribution presented on x-axis with a scale • A “Dot” represents each value in the data set Here all 200 time periods are represented. In larger data sets each dot may represent multiple occurrences of a value. = Minutes • See the Anatomy of Statistics reference sheets for additional examples and discussion.

17. Stem-and-Leaf Plots • Presents all values of the distribution • One column represents the “stems.” • The “leafs” represent the individual values at a stated numeric level (e.g. ones, tenths, etc.) • Must contain a “Key” to identify the level of the stems and leaves. Stem-and-Leaf Plot (single stem) Old Faithful Data presented as a Single Stem-and-Leaf Stem-and-leaf of C1 N = 200 Leaf Unit = 1.0 4 12255558 5 00011111122222233334444555556666677777888889 6 01112223344555666777889 7 012569 8 01111111111112222222222222222333333333334444444444555555555555566+ 9 00112556778 10 1 • See the Anatomy of Statistics reference sheets for additional examples and discussion.

18. Stem-and-Leaf (double stem) • Breaks each stem into two parts (e.g. a stem representing 40-49 becomes 40-44 and 45-49) Old Faithful Data presented as a Double Stem-and-Leaf Stem-and-leaf of C1 N = 200 Leaf Unit = 1.0 4 122 4 55558 5 00011111122222233334444 5 555556666677777888889 6 01112223344 6 555666777889 7 012 7 569 8 01111111111112222222222222222333333333334444444444 8 555555555555566666666666777777777777778888888888888888888 9 00112 9 556778 10 1

20. Create a frequency table containing 9 classes, with the first lower class limit at 0. Identify: • Width of the classes: _______ • Upper Class Limit of the fifth class: ______ • Upper Boundary of the last class: _______ • Midpoint of the third class: ________ • Lower Cutpoint of the sixth class: _______

21. Quantitative Presentations The following data represents the ages of 30 students in a statistics class. Construct a frequency distribution that has five classes. Graphically present these data as a histogram, stem-&-leaf plot and a dot plot. Ages of Students

22. End of SlidesProblem answers follow this slide.

23. Problem Answer: Frequency Table (fruit fly data) Create a frequency table containing 9 classes, with the first lower class limit at 0. Identify: • Width of the classes: 10% • Upper Class Limit of the fifth class: 49 • Upper Boundary of the last class: 89.5 • Midpoint of the third class: 24.5 • Lower Cutpoint of the sixth class: 49.5

24. Problem Answer: Frequency Table & ChartsDistribution (age data)