1. Chapter Four Making and Describing Graphs of Quantitative Variables
Center, Shape. and Spread

3. Describing Data With Graphs
Histograms, Stem and Leaf Plots, Dot Plots

5. Histograms
Histograms are bar charts for QUANTITATVE data where the upper class limit of one class is the same as the lower class limit of the next class. As a result, the bars are next to each other in a histogram.
Dem: Histogram

7. Stem and Leaf Plots
In a stem and leaf plot, data is sorted to reveal the underlying distribution. To construct a stem and leaf plot:
Separate the right most digit from the rest of the digits. This right most digit becomes the leaf.
Insert a legend into the stem and leaf plot indicating the position of this leaf.
The rest of the number without the leaf becomes the stem.
Organize the stems in order, draw a vertical line and place the leaves in order on the other side of the vertical line.

9. Stem and Leaf Plot Example
For the data: 25.6, 25.2, 25.0, 24.1, 24.9, 23.8, 23.6, 23.6, 25.7, 22.0 Organize a stem and leaf plot.

10. Applet for Stem and Leaf Graph

11. Dot Plots
A dot plot consists of a graph in which each data value is plotted as a point (dot) along a scale of values. When values occur more than once, they are plotted as dots stacked vertically above the scale value.

13. Dot Plot Example Give a dot plot for the following values:
201, 201, 200, 203, 203, 203, 205, 205,207

15. Outliers
Outlier – an observation that is unusually large or small relative to the other values in the data set.
Causes of Outliers
a) the measurement is incorrectly observed, recorded, or entered into a computer
b) the measurement comes from a different population
c) the measurement is correct but represents a rare event

16. Numerical Measures
Another way to give information about a sample or a population besides a graph is to use numerical measures about the sample or population.
Numerical measures about a sample are called statistics.
Numerical measures about a population are called parameters.

17. Numerical Measures of Central Tendency
Mean
Mode
Median
Midrange

18. Mean
Arithmetic Mean – the sum of the measurements divided by the total number of measurements.
Population symbol for mean is µ
( µ is a parameter)
The symbol for the mean of a sample is x bar ( x bar is a statistic)

19. Finding the Arithmetic Mean from a Frequency Table
Class
Freq (f)
Mark (x)
200<210 6
205
1230
210<220 10
215
2150
220<230 14
225
3150
Total 30
6530
Mean
6530/30=217.7

20. Round-Off Rule
Carry one more decimal place than is present in the original data set. DO NOT ROUND UNTIL THE END. Let your calculator carry as many decimal places as it can until the very end.

21. Other Measures of Central Tendency
Mode – the most popular ( the data item that occurs the most often)
Median – when data is ranked, the median is the middle piece of data. When there is an even amount of data, the two middle pieces of data are added together and divided by two.
Midrange – the highest plus lowest data entries divided by two. (We will not use this.)

22. Shape of Graphs Symmetric Approximately symmetric
Uniform (or approximately uniform)
Skewed right
Skewed left
Uni-modal
Bi-modal
Gap
Data Far from others
How Class Width Affects a Histogram

23. Skewness
A distribution of data is SKEWED if it is not symmetric and extends more to one side than the other. (A distribution is symmetric if the left half of its histogram is roughly a mirror image of its right half.)

24. Skewness
1) Skewed LEFT (Negatively skewed): The mean and the median are to the left of the mode.
2) Symmetric (Zero Skewness): The mean, median, and the mode are the same.
3) Skewed right (Positively Skewed): The mean and the median are to the right of the mode.
Demo of Skewness and the Mean

25. Spread Spread measures the variation of the data.
(How far apart is the lowest data point from the highest data point?)
Three measures of spread are:
Standard deviation
Variance
Range
We will talk more about spread later.

26. Women versus Men – Study Time
For the followeing data sets, prepare a histogram, dot-plot, and stem and leaf plot. Compare center, shape, and spread for the two sets of data.