2.3 - Organizing and Graphing Quantitative Data
Reasons for Constructing Frequency Distributions:
1. Large data sets can be summarized.
2. We can gain some insight into the nature of data.
3. We have a basis for constructing important graphs.
A Frequency Distribution for Quantitative data lists all the classes and the number of values
that belong to each class. Data presented in the form of a frequency distribution are called grouped
data.
What are some of the issues with a table as the one above?
Make sure that classes do not overlap so that each of the original values must belong to exactly
one class.
Class width = Upper boundary – Lower boundary
Class midpoint or mark =
Lower boundary + Upper boundary
2
Constructing a Frequency Distribution:
1. Decide on the number of classes (should be between 5 and 20). The bigger the data set, the
more classes you should choose. (One way to help you choose number of classes is Sturge’s
formula: # of classes = 1 + 3.3 log n , where n is the number of observations.)
2. Calculate approximate class width. Round to a convenient number.
Class width =
Largest value - Smallest value
Number of classes that you want
3. Starting point: Begin by choosing a lower limit of the first class, which should be less than or
equal to your smallest value.
4. Using the lower limit of the first class and the class width, proceed to list all the classes.
5. Go through the data set putting a tally in the appropriate class for each data value.
Try to use the same width for all classes, although it is sometimes impossible to avoid open-ended
intervals, such as “65 years or older.”
Make sure to include all classes, even those with a frequency of zero.
ex.
How many classes does the frequency distribution have?
What is the class width?
Is it clear where a value belongs?
How big was the sample of pennies?
Could we have omitted the classes with no frequencies,
or do those classes tell us something important?
Randomly Selected Pennies
Weights of Pennies
in grams
2.40-2.49
2.50-2.59
2.60-2.69
2.70-2.79
2.80-2.89
2.90-2.99
3.00-3.09
3.10-3.19
Frequency
18
59
0
0
0
2
25
8
Histograms
A histogram is a graph in which classes are marked on the horizontal axis and the frequencies,
relative frequencies, or percentages are marked on the vertical axis.
The frequencies, relative frequencies, or percentages
are represented by the heights of the bars.
In a histogram, the bars are drawn adjacent to each other
(that is, there is no space in-between them as in a bar graph).
Frequency Histograms vs. Relative Frequency Histograms or Percentage Histograms
A polygon is a graph formed by joining the midpoints of the tops of successive bars in a
histogram with straight lines.
Note that we create one extra class at each end, both of which have zero frequency.
As the number of classes is increased,
the polygon eventually becomes a smooth curve
2.4 - Shapes of Histograms
Symmetric. Bell shaped.
Skewed right.
Symmetric. Bimodal.
Skewed left.
Uniform or Rectangular.
Ex. The following data give the numbers of computer keyboards assembled at the Twentieth
Century Electronics Company for a sample of 25 days.
45 52 48 41 56 46 44 42 48 53 51 53 51
48 46 43 52 50 54 47 44 47 50 49 52
a. Make the frequency distribution table for these data.
b. Calculate the relative frequencies for all classes.
c. Construct a histogram for the relative frequency distribution.
d. Construct a polygon for the relative frequency distribution.