Sample Statistics on TI-83/84

Summary: You can use your TI-83/84 to find measures of central tendency and measures of dispersion for a sample.

See also: MATH200A Program — Basic Statistics Utilities for TI-83/84 gives a downloadable program to plot histograms and box-whisker diagrams.

See also: optional advanced material: MATH200B Program part 1 gives a downloadable program that computes skewness and kurtosis, two numerical measures of shape

Descriptive Statistics for a List of Numbers

Quiz scores in a (fictitious) class were 10.5, 13.5, 8, 12, 11.3, 9, 9.5, 5, 15, 2.5, 10.5, 7, 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. Show your work: write down 1-VarStats and the list name.	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 entering numbers twice.
• mean x̅ = 9.72
  (Write down symbol μ instead of x̅ if this is a population mean.)
• standard deviation s = 3.17
  Since this data set is a sample, use Sx and write s for the standard deviation. (When the data set is the whole population, use σx and write σ for the standard deviation.)
  If rounding is necessary, remember that we round mean and standard deviation to one decimal place more than the data.
• 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 — in this case it’s the five-number summary.

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

You can tell the shape of the distribution. Since the mean x̅ = 9.72 is just a hair less than the median Med or x̃ = 10.5, you know that the distribution is slightly skewed left.

The range is Xmax − Xmin = 15 − 2.5 = 12.5.

The interquartile range or IQR is Q3 − Q1 = 11.5 − 8 = 3.5. Recall that we use 1.5 × IQR to classify outliers: we call a data point an outlier if it’s at least that far below Q1 or above Q3.

In this case 1.5 × IQR = 1.5 × 3.5 = 5.25, Q1 − 5.25 = 2.75, and Q3 + 5.25 = 16.75, so we can say that any data points below 2.75 or above 16.75 are outliers. (Making a box-whisker plot is easier: see MATH200A Program part 2.)

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 = 3.17² = 10.05 — see the Big No-no for the reason. You could enter 3.165257832², 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 if your data set is a sample or σx if your data set is the whole population.	[3] for Sx or [4] for σx.
Square it.   The variance is s² = 10.02. (If the data set was a whole population, you’d use σ² for the variance.)	[x²] [ENTER]

Descriptive Statistics for a Frequency Distribution

The grouped frequency distribution at right is the ages reported by Roman Catholic nuns, from Johnson & Kuby 2004 [full citation at https://BrownMath.com/swt/sources.htm#so_Johnson2004], 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. Show your work: write down 1-VarStats and both lists.   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. If you see n is the number of classes instead of the number of data points, redo your 1-VarStats and this time specify both lists.

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 x̅ = 63.9
  (Write symbol μ if this is a population mean.)
• standard deviation s = 15.4
  If this data set is a sample, use Sx and write s for the standard deviation; if this data set is the whole population (including a probability distribution), use σx and write 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 classes that actually contain those statistics.

As a general rule, the five-number summary from a grouped frequency distribution is not worth reporting. The numbers will be only approximations, because the calculator has only the class midpoints to work with and not the original data.

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 if your data set is a sample or σx if your data set is the whole population.	[3] for Sx or [4] for σx.
Square it.   The variance is s² = 238.2. (If the data set was a whole population, the variance would be σ².)	[x²] [ENTER]