# Frequency Distributions with a TI-83+ Calculator

The amount of protein (in grams) for a selection of fast-food sandwiches is given below.

23, 30, 20, 27, 44, 26, 35, 20, 29, 29,
25, 15, 18, 27, 19, 22, 12, 26, 34, 15,
27, 35, 26, 43, 35, 14, 24, 12, 23, 31,
40, 35, 38, 57, 22, 42, 24, 21, 27, 33


Suppose we wish to construct a frequency distribution for this data (using a TI-83+ calculator).

First, we need to get the values into our calculator...

• Hit the STAT button and select EDIT:!ClrList. Now hit the (2nd)L1 button, followed by the ENTER button to clear any previous data in the list L1.
• Now hit the STAT button again, and select EDIT:Edit... followed by the ENTER button. The list editing screen should pop up.
• Use the arrow keys to move the highlighted cell to the first entry of the list L1, if it is not already there.
• Begin typing the values above, hitting the ENTER key after each value to add it to the list.
• When you are done, double check your entries! (You can scroll up and down the list with the arrow keys.) It is VERY easy to make a mistake typing here, and you won't necessarily catch it later -- so look closely now!
• When you are done, hit (2nd)QUIT to return to the home screen.

Now, our data would be much easier to work with if it was sorted from lowest to highest...

• Hit the STAT button and select EDIT:SortA(. Then, finish up the command by hitting the (2nd)L1 button and then the ) button, and finally the ENTER button.
• You can see the fruits of your labors by hitting going back to the list editing screen (STAT, EDIT:Edit..., ENTER) They should all be in order now from smallest to largest.

We need to know our maximum and minimum data values. There are two ways to do this.

1. Scroll through the list using the arrow keys in the list editing screen.
2. Hit the STAT button, and then select CALC:1-Var Stats followed by the ENTER button. Now scroll down to the bottom of the statistics produced, using the down arrow key. The minimum and maximum data values in list L1 will be named as minX and maxX, respectively.

For the data above, we have a minimum of 12 and a maximum of 57.

We can use these as our minimum and maximum class limits, but round numbers are nice and pretty looking, so you may want to consider lowering your minimum class limit (Just don't stray too far away from the actual minimum!)

In our case, 12 is fairly close to a nice round 10, so let's make that our minimum class limit.

No data values should fall on the borders of our classes. An easy way to stop that from happening is to take your minimum class limit and subtract either .5, .05, .005, or something similar -- so that the number of decimal places of your answer is one greater than the number of decimal places your most precise data has.

For this example, that would make the lowest class boundary 9.5.

There is a slight difference between a "class limit" and a "class boundary".

• A class limit is the smallest or largest data value that would fall into a particular class at a given level of precision (like values with no decimal places, for instance).
• A class boundary doesn't care about the number of decimal places. Anything immediately above it falls into one class, and anything immediately below it falls into another class.

Decide how many classes you want, if this is not already known. You will probably need between 5 and 20 classes, with more classes being more appropriate for larger data sets.

Suppose, for this example, that we want 10 classes.

Now, we need to find out how wide to make each class.

• Subtract your minimum class limit from your maximum, and divide the result by the number of classes you want.

For our example, this calculation yields:

$$\frac{57-10}{10} = 4.7$$

Now, round this up to the next value with the same precision as your data.

In our example, this would mean rounding 4.7 up to the next integer, which is 5. This is our class width.

We can now start describing our classes.

Our class boundaries start at 9.5 and have width 5, so we have:

 9.5 - 14.5
14.5 - 19.5
19.5 - 24.5
24.5 - 29.5
29.5 - 34.5
34.5 - 39.5
39.5 - 44.5
44.5 - 49.5
49.5 - 54.5
54.5 - 59.5


Our class limits only include values at the same precision as our data (in our case, integers), so we "tighten" these class boundary intervals up a bit when describing the class limts:

Class Boundaries  Class Limits
9.5 - 14.5         10-14
14.5 - 19.5         15-19
19.5 - 24.5         20-24
24.5 - 29.5         25-29
29.5 - 34.5         30-34
34.5 - 39.5         35-39
39.5 - 44.5         40-44
44.5 - 49.5         45-49
49.5 - 54.5         50-54
54.5 - 59.5         55-59


Now its just a matter of scanning through our data and marking how many fall in each class. (This is where having things sorted pays off!) Use the list editor to do this. (STAT, EDIT:Edit..., ENTER)

Class Boundaries  Class Limits   Frequency
9.5 - 14.5         10-14            3
14.5 - 19.5         15-19            4
19.5 - 24.5         20-24            9
24.5 - 29.5         25-29           10
29.5 - 34.5         30-34            4
34.5 - 39.5         35-39            5
39.5 - 44.5         40-44            4
44.5 - 49.5         45-49            0
49.5 - 54.5         50-54            0
54.5 - 59.5         55-59            1


Depending on what information you need, you may also want to list the relative frequencies for each class. This is easy to add to our table.

First count how many data values you have (which we call n).

(A quick way to do this is to use STAT, CALC:1-Var Stats, ENTER, ENTER, and look at the value of n).

Now divide each frequency by n, to find the relative frequency for the corresponding class:

                                            Relative
Class Boundaries  Class Limits   Frequency  Frequency
9.5 - 14.5         10-14            3         .075
14.5 - 19.5         15-19            4         .100
19.5 - 24.5         20-24            9         .225
24.5 - 29.5         25-29           10         .250
29.5 - 34.5         30-34            4         .100
34.5 - 39.5         35-39            5         .125
39.5 - 44.5         40-44            4         .100
44.5 - 49.5         45-49            0         .000
49.5 - 54.5         50-54            0         .000
54.5 - 59.5         55-59            1         .025


If we want to add a column for cumulative frequency, this too is an easy addition.

Simply keep a running total of the relative frequencies from top to bottom, adding the next relative frequency to the last cumulative frequency at each step.

                                            Relative    Cumulative
Class Boundaries  Class Limits   Frequency  Frequency   Frequency
9.5 - 14.5         10-14            3         .075        .075
14.5 - 19.5         15-19            4         .100        .175
19.5 - 24.5         20-24            9         .225        .400
24.5 - 29.5         25-29           10         .250        .650
29.5 - 34.5         30-34            4         .100        .750
34.5 - 39.5         35-39            5         .125        .875
39.5 - 44.5         40-44            4         .100        .975
44.5 - 49.5         45-49            0         .000        .975
49.5 - 54.5         50-54            0         .000        .975
54.5 - 59.5         55-59            1         .025       1.000