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 Data Know 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 cumulative frequencies, 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.
Using the Old Faithful frequency table:
Lower Class Limits – 40, 50, 60, 70, 80, 90, 100
Upper Class Limits – 49, 59, 69, 79, 89, 99, 109
To obtain class boundaries –
1. Find the size of the gap between the upper class limit of one class, and the lower class limit of the next class.
2. Add half that amount to each upper class limit to find the upper class boundaries.
3. Subtract half that amount from each lower class limit to find the lower class boundaries.
50-49=1, ½=.5, =39.5, =49.5
Class Boundaries are – 39.5, 49.5, 59.5, 69.5, 79.5, 89.5, 99.5, 109.5
Class Midpoints – add the lower class limit to the upper class limit and divide the sum by 2.
(40+49)/2=44.5, the midpoint of the first class.
Class Width – using lower class limits: 50-40=10
using lower class boundaries: =10

7. Quantitative Frequency Distributions
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
Grouped Frequency Distributions include a series of consecutive values into a Class (grouping)
Discrete or Continuous data may be presented

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

9. Single-Value Grouped Data Table
Each value is represented as a class.
Note that each class has only one value, as opposed to an interval of values.

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).
Let’s look at a histogram of the Old Faithful data.
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.
Dotplot - each dot represents one observation. For example, on two occasions in this sample, the length of time between eruptions was 90 minutes.
SPSS does not do Dotplots.
= Minutes
See the Anatomy of Statistics reference sheets for additional examples and discussion.

16. Dotplot - each dot represents one observation (here, one fruitfly)
For example 2 of the 125 fruitflies spent 40% of the day sleeping.

17. Stem-and-Leaf Plot (single stem)
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 C N = 200
Leaf Unit = 1.0
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 becomes and 45-49)
Old Faithful Data presented as a Double Stem-and-Leaf
Stem-and-leaf of C N = 200
Leaf Unit = 1.0
4 122
7 012
7 569
10 1
Double Stem-and Leaf

19. Single Stem-and-Leaf -
Ordered versus Non-ordered
Depth -

20. Width of the classes: _______
Percent of Day Spent Sleeping by Male Fruitflies 1 8 12 17 20 23 28 35 66 2 9 13 24 36 71 4 29 73 14 21 37 81 25 83 5 10 22 30 38 6 18 26 31 40 15 27 32 42 43 33 49 34 7 16 19 50 62
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 18 20 21 27 29 19 30 32 34 24 37 38 22 39 44 33 46 54 49 51

22. End of Slides Problem 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
Chart here is a frequency distribution table. It is also referred to by Weiss as a grouped data table.
Each observation must belong to one and only one class.
All classes should be of the same width.
Lower Cutpoint - the smallest value that can go in a class. In the 30<40 class, 30 is the lower cutpoint.
Upper Cutpoint - the smallest value that can go in the next higher class. The upper cutpoint of a class is the same as the lower cutpoint of the next higher class. In the 30<40 class, 40 is the upper cutpoint.
Midpoint - the middle of a class, obtained by adding the lower and upper cutpoints together, and then dividing by 2 (the average). In the 30<40 class, (30+40)/2 = 35
Width - The difference between the upper and lower cutpoints of a class. Using the 30<40 class = 10

24. Problem Answer: Frequency Table & ChartsDistribution (age data)
Age of Students Stem-and-Leaf Plot
Frequency Stem & Leaf
Stem width:
Each leaf: case(s)