Descriptive Statistics of a Data Set on the TI-83/84

Summary:  You can use your TI-83/84 to find measures of central tendency and measures of dispersion for a data set. You can also use it to make a box-whisker diagram to illustrate the five-number summary.

See also:  Histograms of Numeric Data on the TI-83/84
optional advanced material: Skewness and Kurtosis on the TI-83/84

Descriptive Statistics for a List of Numbers

Quiz scores in a (fictitious) class were 10.5, 6, 8, 6, 11.3, 9, 9, 5, 3.5, 1, 1, 6.8, 11.5, 10, and 10.5. It’s hard to get much of a sense of the class by just staring at the numbers, but you can easily compute the common measures of center and spread by using your TI-83 or TI-84.

Step 1: Enter the numbers in L1

By the way, this note uses list L1, but you can actually use any list you like, as long as you enter the actual list name in the 1-Var Stats command in Step 2. (It doesn’t matter whether there are numbers in any other list.)

Enter the data points.

[STAT] [1] selects the list-edit screen.   Cursor onto the label L1 at top of first column, then [CLEAR] [ENTER] erases the list. Enter the x values.

Step 2: Compute the Statistics

Select the 1-Var Stats command.	[STAT] [►] [1] pastes the command to the home screen.
Specify which statistics list contains the data set.	Assuming you used L1, enter [2nd 1 makes L1].   Press [ENTER] to execute the command.

The important statistics are

• sample size n = 15
  Always check this first to guard against leaving out numbers or enterng numbers twice.
• mean x̄ = 7.27
  (Use symbol μ if this is a population mean.)
• standard deviation s = 3.49
  If this data set is a sample, use S_x or s for the standard deviation; if this data set is the whole population, use σ_x or σ for the standard deviation.
• variance is not shown on this screen; see Step 3 below.

The down arrow on the screen tells you that there’s more information if you scroll down.   You can tell the shape of the distribution. Since the mean x̄ = 7.28 is less than the median x̃ = 8, you know that the distribution is skewed left.

[▼ 5 times] for the five-number summary.

Step 3: Find the Variance

Your TI-83 or TI-84 doesn’t find the variance for you automatically, but since the standard deviation is the square root of the variance, you can find the variance by squaring the standard deviation.

It would be wrong to compute s² = 3.49² = 12.18 — see The Big No-no for the reason. You could enter 3.488852997², but that’s tedious and error prone, as well as being overkill.

Instead, use the value that the calculator has stored in a variable.

Select statistics variables.	[VARS] [5]
Select the correct standard deviation: Sx for a sample or σx for a population.	[3] for Sx or [4] for σx.
Square it.   The variance is s² = 12.17	[x²] [ENTER]

Step 4: Make a Box-Whisker Plot

You already have the data in L1, so all you have to do is set up the Stat Plot screen. Fortunately, there’s no need to mess with the Window screen, because a ZOOM command will adjust the window nicely.

Turn off other plots.	Press [Y=]. Cursor to each highlighted = sign or Plot number and press [ENTER] to deactivate.
Turn on Stat Plot 1 as a modified box-whisker diagram.	[2nd Y= makes STAT PLOT] [1] [ENTER]
Select the modified box-whisker plot, which is the first one in the second row.	[▼] [► 3 times] [ENTER]
Specify L1 and a frequency of 1.	[▼] [2nd 1 makes L1]   [▼] then [ALPHA] to turn off alpha mode, followed by [1]
Select a square or cross for plotting any outliers.	[▼], [►] if you wish, then [ENTER]
Display the plot by zooming automatically to the statistics.   The plot shows that the distribution is skewed left. (You already knew that because x̄ <x̃, but it’s much easier to see in a box-whisker plot or a histogram.)   The plot shows that there are no outliers in this data set.	[ZOOM] [9] executes the ZoomStat command, which adjusts the window according to the numbers that you’re plotting.

By the way, you can press the [TRACE] key and then use the left and right arrows to reveal the numeric values of the minimum, Q1, x̃, Q3, and maximum, as well as any outliers.

Descriptive Statistics for a Frequency Distribution

The grouped frequency distribution at right is the ages reported by Roman Catholic nuns, from Johnson & Kuby, Elementary Statistics 9/e (Thomson, 2004), page 67. Let’s use the TI-83/84 to compute statistics.

Step 1: Enter class marks in L1 and frequencies in L2

By the way, this note uses L1 and L2, but you can use any lists you like, as long as you enter the actual list names in the 1-Var Stats command in Step 2. (It doesn’t matter whether there are numbers in any other list.)

This example is for a grouped frequency distribution. If you have an ungrouped frequency distribution, you can compute statistics in the same way. The only difference is that your first list will contain the actual values instead of the class marks.

Enter the class marks in L1.   (The class mark is the midpoint of each class.)	[STAT] [1] selects the list-edit screen.   Cursor onto the label L1 at top of first column, then [CLEAR] [ENTER] erases the list. Enter the class marks.   (If you have only the class boundaries, you can make the TI-83/84 do the work for you. It will compute the class marks automatically if you enter the class boundaries in the form (20+30)÷2.)
Enter the frequencies in L2.	Cursor onto the label L2 at top of first column, then [CLEAR] [ENTER] erases the list. Enter the frequencies.

Step 2: Compute the Statistics

Select the 1-Var Stats command.	[STAT] [►] [1] pastes the command to the home screen.
Specify which statistics lists contain the data set and the frequencies, in that order.   Important: You must supply both lists. That’s the only way the calculator knows you have a frequency distribution. Always check the sample size n in the output, to guard against forgetting to enter the second list.	Assuming you used L1 and L2, enter [2nd 1 makes L1] [,] [2nd 2 makes L2].   Press [ENTER] to execute the command.

The important statistics are

• sample size n = 997
  Again, if this is a low number it means you forgot to specify frequencies on the 1-Var Stats command.
• mean x̄ = 63.9
  (Use symbol μ if this is a population mean.)
• standard deviation s = 15.4
  If this data set is a sample, use S_x or s for the standard deviation; if this data set is the whole population (including a probability distribution), use σ_x or σ for the standard deviation.
• variance is not shown on this screen; see Step 3 below.

Remember that the values on this screen are approximate because the frequency distribution is an approximation of the original raw data. For most real-life data sets, the approximation is quite good, and it is very good for moderate to large data sets.

The down arrow on the screen tells you that there’s more information if you scroll down. However, since the numbers you enter in a grouped frequency distribution are only approximate, the five-number summary is only approximate. The Min and Max are just the highest and lowest classes. Q1, Med, and Q3 are at best the midpoints of the clases that actually contain those statistics.

As a general rule, the five-number summary from a grouped frequency distribution is not worth reporting.

The box-whisker plot is a picture of the five-number summary. Therefore, don’t draw a box-whisker plot of a grouped frequency distribution.

Step 3: Find the Variance

Just as with a simple list of numbers, you find the variance by squaring the standard deviation.

It would be wrong to compute s² = 15.4² = 237.2 — see The Big No-no for the reason.

Instead, use the value that the calculator has stored in a variable.

Select statistics variables.	[VARS] [5]
Select the correct standard deviation: Sx for a sample or σx for a population.	[3] for Sx or [4] for σx.
Square it.   The variance is s² = 238.2	[x²] [ENTER]

Step 4: Make a Box-Whisker Plot

If you have an ungrouped frequency distribution, you can make a box-whisker plot using the directions above for a list of numbers. Just answer the Freq: prompt with the list that contains the frequencies. This doesn’t make sense for a grouped frequency distribution, however.

Compare Data Sets with Box-Whisker Diagrams

Sullivan, Michael, Fundamentals of Statistics (Pearson Prentice Hall, 2008), page 163, shows data for two groups of rats. One group was sent into space; the control group was treated the same except for the space flight. Their red blood cell mass was measured in milliliters.

Let’s draw the boxplots on the same scale.

Enter the flight group in L1 and the control group in L2. (You could use any lists, but we’ll use L1 and L2.)
You already have the plot set up for the first group. If not, please review Step 4 above.   Now set up the second plot.	[2nd Y= makes STAT PLOT] [2] [ENTER] turns on the plot.   [▼] [► 3 times] [ENTER] selects a modified box plot.   [▼] [2nd 2 makes L2] selects statistics list 2.   [▼] [ALPHA] [1] enters 1 for frequency. Select box or cross for any outliers.
Order the calculator to plot both groups on the same scale.   You can see that the flight group had lower blood mass overall, and it was skewed right, meaning that most of the rats had lower scores.   There were no outliers in either group.	[ZOOM] [9]
You can explore the five-number summaries by tracing the plots.	[TRACE] and notice that the upper left of the screen shows P1:L1. This tells you that you’re tracing the flight group, because you put its numbers in list 1. Use the left and right arrows to explore the five numbers.   To switch between groups (lists), use the up or down arrow. The legend in the upper left corner of the screen always tells you which group you’re tracing.