Descriptive Statistics Utilities for TI-83/84

See also:  Troubles? See TI-83/84 Troubleshooting.

MATH200D Program Overview

The Descriptive Statistics Utilities program, MATH200D, doesn’t do anything your calculator can’t already do, but it greatly simplifies the procedures. Check your class requirements to determine which of these utilities you’ll need.

Program change:  At the end of the Fall 2009 semester, this program and MATH200I will be withdrawn, replaced by MATH200 Utilities for TI-83/84 and Extra Statistics Utilities for TI-83/84. This will be simpler for TC3 students, with all the required utilities in one program and all the optional utilities in the other.

Getting the Program

There are three methods to get the program into your calculator:

• If a classmate has the program on her calculator (any model TI-83/84), she can transfer it to yours using the short cable with headphone-style plugs that comes with all TI calculators. On your calculator, press [2nd x,T,θ,n makes LINK] [►] [ENTER], and then on hers press [2nd x,T,θ,n makes LINK] [3], select MATH200D, then press [►] [ENTER].
• Or, download MATH200D.ZIP (61 KB, updated 24 Aug 2009), unzip it, and transfer file MATH200D.8XP to your calculator. (This requires TI-Connect or TI-Graph Link software software and a cable.)
• Or, as a last resort, key in the program. See MATH200D.PDF and MATH200D_HINTS.HTM in the MATH200D.ZIP file.

Special Note for Original TI-83

If you have a TI-83 Plus or Silver Edition, the above program version is fine for you and you should ignore this Special Note. But if you have the original TI-83, without a Plus or Silver designation, then this note applies to you.

Instead of the MATH200D program, you need M200D83. If transferring it from a colleague’s calculator, check the program name carefully. If you’re getting the program from the MATH200D.ZIP file, you need MATH200D_for_original_TI83.83P.

The two versions are functionally identical, but the “original TI-83” version uses all capital letters for prompts and displays because the original TI-83 couldn’t handle lower case in programs. (ρ and σ are replaced with RHO and σx for the same reason.) This Web page shows all screens from the TI-83 Plus or TI-84 version, because most students have a calculator that can handle it.

Using the Program

Press the [PRGM] key. If you can see MATH200D in the menu, press its number; otherwise, scroll to it and press [ENTER]. When the program name appears on your home screen, press [ENTER] to run it.

The menu at right shows what the program can do:

1. Histograms etc: make a histogram or polygon, or overlay both, for a frequency distribution, a relative frequency distribution, a probability distribution, or a simple list of numbers.
2. Box-whisker: plot a box-whisker diagram, showing any outliers
3. Time series: plot time-series data
4. Skew/kurtosis: compute skewness and kurtosis, which are numerical measures of the shape of a distribution
5. Binomial prob: compute probability or cumulative probability of a binomial distribution
6. Binom PD histo: plot a histogram of a binomial distribution
7. Normality chk: test whether sample is drawn from a normal distribution

Each procedure leaves its results in variables in case you want to use them for further computations. Please see the reference section below.

If you should ever need to break out of the program before finishing the prompts, press [ON] [1].

1. Histograms and Frequency Polygons

To use the program, put the class marks or data in a statistics list and the frequencies (if applicable) in another. (Your book might use the term class midpoints instead of class marks; they mean the same thing.)

Then press [PROG], select MATH200D, and press 1:Histograms etc. The program will prompt you for the necessary information and will check silently to make sure your inputs are in valid form. Then it will ask whether you want a histogram, polygon, or both, and will produce your desired graph. (The program uses an algorithm to ensure that there are an appropriate number of dots vertically on the screen.)

Restrictions: If you have frequency data, the data list and frequency list must be the same length and the class widths in the data list must all be the same; the program checks this. The TI-83 and TI-84 won’t let you make a histogram with more than 47 classes, and the program also checks this. Finally, the program also won’t let you make a grouped frequency histogram with just one or two classes, because that’s silly.

Grouped Frequency Data

The grouped frequency distribution at right shows the ages reported by Roman Catholic nuns, from Johnson & Kuby, Elementary Statistics 9/e (Thomson, 2004), page 67. Show the data as a histogram and as a frequency polygon.

Solution: Your class marks (class midpoints) are 25, 35, up through 85, and your class width is 10. On the STAT EDIT screen, enter those class marks in one list and the frequencies in another. Here the class marks have been entered in L5 and the frequencies in L6.

Run the MATH200D program and select 1:Histograms etc. Specify your list of class marks. On the Data Arrangement screen, select [2] for grouped frequency distribution. (Class marks must be equally spaced, and the program will verify that.)

Next, specify your frequency list, and then select which plots you want. In the illustration, I’m selecting both plots. If you’re not doing frequency polygons in your class, select [1] for just the histogram.

The output is shown at right, once with the histogram and polygon on the same screen, and once with just the histogram.

optional extra: Tracing the histogram or polygon

You can trace the histogram by pressing [TRACE]. This lets you see the class boundaries and number of data points in each class.

Press [◄] and [►] to move through the classes. To suppress the tracing information, press [GRAPH] again.

To trace the polygon, if you selected both, press [TRACE] [▼]. This lets you see the class marks and number of data points in each class, instead of the class boundaries.

Why press [▼]? When you press [TRACE], the calculator starts with a trace of Stat Plot 1. The up or down cursor key moves between plots. Since the frequency polygon is Stat Plot 2, you are tracing it when you see P2 in the upper left corner, as shown in the illustration.

List of Numbers

Suppose you want to make a histogram of the class performance on a 15-point quiz, where the scores are shown at left. You don’t want to bother to group these 19 scores by hand, so you enter them in a statistics list such as L1 and let the program do the grouping for you. (An alternative graphical display is the box-whisker plot.)

Run the MATH200D program and select 1:Histograms etc, then enter the name of the list that contains your numbers. When the program asks your data arrangement, select [1] for a plain list of numbers.

For a plain list of numbers, the program needs to know how you want to group your data, so it asks you to specify the lower bound of the first class as well as the class width. Since 0 is the lowest possible grade, that’s the obvious lower bound for this example. The quiz has a possible maximum score of 15, and 10% of that is 1.5 points. This is a good class width because D, C, B, and A will each be one bar of the histogram.

At this point you may see a pause as the program computes the number of classes and places each data point into a class. (It uses statistics lists LD for the class marks (class midpoints) and LF for the computed frequencies.)

For a simple list of numbers, the program will make only a histogram, not a frequency polygon.

optional extra: Tracing the histogram

In first looking at that histogram, you might think there are four Ds, three Cs, two Bs, and one A. But when you check this by pressing [TRACE] you see that’s not correct. The highest class is the 15 to 16.5 class, since someone had a perfect score of 15. (Remember that when a value is right on a class boundary, it is always assigned to the higher class.) So the top two classes in the histogram represent As (three students), the next lower is the three Bs, the next lower (shown at right) is four Cs, the next lower is two Ds, and the rest are Fs.

Ungrouped Frequency Data

You have recorded the numbers of children in 30 randomly selected families that used a community center in a given week, and you want to show a picture of the discrete distribution. There are only a few different values (numbers of children per family), so you choose an ungrouped frequency distribution.

Enter the data in two lists such as L3 and L4, run the MATH200D program, and select 1:Histograms etc. Enter your data list as usual. For Data Arrangement, select [3], ungrouped distribution.

When prompted, enter the frequency list, then select whether you want a histogram, a frequency polygon, or both. I’ve selected a histogram, and the results are shown below.

The vertical line in the histogram is the y axis — you can remove it, if you want, by pressing [2nd ZOOM makes FORMAT] and selecting AXES OFF. Also in the histogram, notice that the vertical rows of dots run through the center of each bar rather than along the edges. This reminds you that you should label the bars of an ungrouped frequency distribution under the centers, not the edges as you would label a grouped histogram.

If you want to trace the ungrouped frequency histogram, follow the same procedure as for tracing a grouped frequency histogram.

Relative Frequencies or Probabilities

If you have a relative frequency distribution or probability distribution, you plot it in almost the same way as a frequency distribution. The main difference is that the relative frequencies or probabilities must add up to 1, and the program checks this for you.

(There’s a special case: the binomial probability distribution. For this, please see Histogram of a Binomial Distribution below.)

Here’s an example of a general discrete probability distribution, drawn from the rainy-day game Yahtzee. In Yahtzee you roll five dice and try to make various combinations. Shown at right are the probabilities for number of dice alike when you roll five standard six-sided dice. (Thanks to Paul Sperry for help with the probabilities.) You can make a histogram of this probability distribution.

Notice, by the way, that you’re more than twice as likely to roll three of a kind as to roll “none of a kind” or all five different: P(3) = 1500/7776, and P(1) = 720/7776. And you’re over seven times as likely to roll two of a kind (either two the same and three all different, or two pairs with the fifth die different): P(2) = 5400/7776. Of course, Yahtzee isn’t just about the initial roll. You get two tries to improve your combinations by re-rolling some of your dice.

As before, put the x’s (1–5 this time) in one list and the p’s in another. Run the program and select 1:Histograms etc, then specify your data list. The data arrangement this time is a probability distribution, so specify [4] and then enter your probability list.

The finished probability histogram is shown at right. (For a probability distribution, the program automatically makes a histogram, with no option to make a polygon.) Notice that the bar for two of a kind is much higher than any of the others.

If it wasn’t obvious from the numbers, you can see from the histogram that this distribution is skewed right. If you like, you can press the [TRACE] button and see the numerical value of the probability for each outcome. For example, P(2) = 0.6944, meaning that when you roll five dice you have almost a 70% chance of getting two of a kind.

What about two of a kind or better? This is a classic “at least” problem, and the complement is your friend. P(1) is 0.0926 and 1−P(1) = 0.9074. You have better than a 90% chance of getting at least two of a kind when rolling five dice.

2. Box-Whisker Plot Showing Outliers

Summary:  A modified box-whisker plot is a quick graphical representation of a data set. It plots the five-number summary (minimum, first quartile, median, third quartile, maximum) and also shows outliers if the data set contains any. The 2:Box-whisker part of the MATH200D program makes a modified box-whisker diagram for one data set or compares two or three data sets by stacking box-whisker plots.

One Sample, List of Numbers

Here again is the set of quiz scores. You’ve already seen them graphed as a histogram, but a box-whisker plot is another way to get a sense of the shape of the data.

Enter the data in any statistics list and run the MATH200D program. Select 2:Box-whisker and you see a prompt for the number of samples. You can make a box-whisker plot of a single data set, or you can compare two or three data sets. The quiz scores are a single sample, so you choose [1].

The program asks you for the data list, and then whether you have a plain list of numbers or an ungrouped frequency distribution. Here you have a plain list of numbers, so you choose data arrangement [1].

Caution: Never make a box-whisker of a grouped frequency distribution. Only a simple list of numbers or an ungrouped frequency distribution is suitable for a box-whisker plot. If you have a grouped frequency distribution, a box-whisker plot won’t be accurate and you should be using a histogram.

The box-whisker plot now appears (below left). You can see at a glance that it has no outliers, and that it’s slightly skewed left.

You can also trace the box-whisker, to see the five-number summary and the values of any outliers. Press the [TRACE] key and then use [◄] and [►] to move left and right. In the illustration below right you can see that the median quiz score was 10.5.

What would an outlier look like on a box-whisker plot? Any outliers show up as isolated points separate from the main diagram.

For example, take the same set of data, but append a twentieth data point, 22. Now make the plot and you’ll see the result at right. The [TRACE] key and arrow keys will display the values of the outliers as well as the five-number summary.

Stacked Boxplots to Compare Data Sets

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.

Plotting the two data sets as two boxplots on the same screen is a good way to get a sense of whether there is a difference between the samples. (Later in the course, you’ll learn how to use a two-sample t test to tell whether there is a difference between the populations of all space rats and all earthbound rats.)

Put the flight group in one statistics list and the control group in another. Then run the MATH200D program, select 2:Box-whisker, and select 2:Compare 2 smpl. Enter the names of the two lists, as shown at right.

The results are shown at right. You can see that the flight group, as a group, had lower blood-cell mass than the control group, even though some individuals from the two groups had equal blood-cell mass. Look particularly at the medians: the median for the control group is about equal to the third quartile of the flight group, meaning that about three quarters of the flight rats had blood-cell mass lower than the median of the control group.

3. Time Series Plots

Summary:  To plot a time series or trend line, put the numbers in a statistics list and use the 3:Time series part of the MATH200D program.

Example: Let’s plot the closing prices of Cisco Systems stock over a two-year period. The following table is adapted from Sullivan, Michael, Fundamentals of Statistics (Pearson Prentice Hall, 2008), page 82, which credits NASDAQ as the source.

Enter the closing prices in a statistics list such as L1, ignoring the dates.

Now run the MATH200D program and select 3:Time series. The program prompts you for the data list. (Caution: The program assumes the time intervals are all equal. If they aren’t, the horizontal scale will not be uniform and the graph will not be correct.)

It’s usually good practice to start the vertical scale at zero, or in other words to show the x axis at its proper level on the graph. But the program gives you the choice. If you have good reason, you can let the program scale the data to take up the entire screen. This exaggerates the amount of change from one time period to the next. (If the data include any negative or zero values, the x axis will naturally appear in the graph, and program skips the yes/no prompt.)

Below you see the effect of a “yes” at left and the effect of a “no” at right.

As you can see, the graph that doesn’t include the zero looks a lot more dramatic, with bigger changes. But that can be deceptive. A more accurate picture is shown in the first graph, the one that does include the x axis.

optional extra: Tracing the Graph

If you wish, you can press the [TRACE] key and display the closing prices, scrolling back and forth with the [◄] and [►] keys. If you want to jump to a particular month, say June 2004, the 16th month, type 16 and then press [ENTER].

4. Skewness and Kurtosis

Using the Program

If you have a frequency or probability distribution, put the data points or class marks in one statistics list and the frequencies or probabilities in another. If you have a simple list of numbers, put them in a statistics list.

Then press [PRGM], scroll if necessary and select MATH200D, and in the program menu select 4:Skew/kurtosis. Enter your data list, specify your data arrangement, and if appropriate enter your frequency or probability list. The program will produce a great many statistics.

Example: College Students’ Heights

Here are grouped data for heights of 100 randomly selected male students:

A histogram shows the data are skewed left, not symmetric. But how highly skewed are they? And how does the central peak compare to the normal distribution for height and sharpness? To answer these questions, you have to compute the skewness and kurtosis.

Enter the x’s in one statistics list and the f’s in another. If you’re not sure how to create statistics lists, please see Sample Statistics on TI-83/84.

Then run the MATH200D program and select 4:Skew/kurtosis. Your data arrangement is 2:Freq/prob dist. When prompted, enter the list that contains the x’s and then the list that contains the f’s. I’ve used L5 and L6, but you could use any lists.

The program gives its results on three screens of data.

The first screen shows some basic statistics: the sample size, the mean, the standard deviation, and the variance. M and V are not the proper symbols for mean and variance, as you know, but numeric variables on the TI-83/84 can have only single-letter names. The program stores key results in variables in case you want to do any further computations with them. (Standard deviation isn’t stored in a variable because you can always get it by √V.)

See below for a complete list of variables computed by the program.

The second screen shows results for skewness. The third moment divided by the 1.5 power of the variance is the skewness, which is about −0.11 for this data set. The distribution is therefore negatively skewed (skewed to the left), but can you say that the population is also negatively skewed? To answer that question, use the standard error of skewness, which is also shown on the screen.

As a rule of thumb, if sample skewness is more than about two standard errors either side of zero, you can say that the population is skewed in that direction. In this example, the standard error of skewness is 0.24, and the skewness divided by standard error is shown as −0.45. That last number tells you that skewness is only 0.45 standard errors below zero, so you can’t say whether the population is skewed or symmetric.

The third screen shows results for kurtosis. The fourth moment divided by the square of the variance gives the kurtosis, which is 2.74, so the sample is platykurtic. Comparing this to the “standard” kurtosis of 3, you see that the central peak is a little flatter and less distinct than the central peak of a normal distribution (bell curve). (A kurtosis greater than 3 would mean that the distribution was leptokurtic, with a shorter and higher peak than a bell curve.)

Some authors prefer to consider the excess kurtosis, which is kurtosis minus 3. A bell curve has excess kurtosis of 0, so the excess kurtosis lets you compare the “peakyness” of a distribution to zero to determine whether it’s more or less “peaky” than the normal distribution.

What can you say about the kurtosis of the population from which the sample was taken? The rule of thumb is that an excess kurtosis of at least two standard errors is significant. The standard error of kurtosis is 0.48, and the excess kurtosis is only 0.54 standard errors below zero. (Or, the kurtosis is only 0.54 standard errors below 3.) Therefore you can’t say whether the population is peaked like a normal distribution, more than normal, or less than normal.

Example: Throwing Dice

You can also use this part of the program to compute the shape of a probability distribution. For instance, here's the probability distribution for the number of spots showing when you throw two dice:

The x’s go in one list and the P’s in another. (Enter the probabilities as fractions, not decimals, to ensure that they add to exactly 1. The calculator displays rounded decimals but keeps full precision internally, and the program will tell you if your probabilities don’t add to 1.) Now run the MATH200D program and select 4:Skew/kurtosis, and you’ll see the following results:

On the first screen, note that the sample size n is 1; this is your confirmation that you have a probability distribution.

On the second screen, the skewness is essentially zero. This confirms what you can see in the histogram: the distribution is symmetric. Standard error and test statistic don’t apply because you have a probability distribution (population) rather than a sample.

On the third screen, the kurtosis is 2.37 and the excess kurtosis is −0.63; the dice make a platykurtic distribution. Compared to a normal distribution, this distribution of dice throwing has a lower, less distinct peak and shorter tails.

5. Probability of a Binomial Experiment

Binomial Probability for One x Value

The program always asks you the number of trials n, the probability p of success on one trial, and the number of successes from and to. If you want a specific number of successes rather than a range, then from and to will be the same number.

Example 1: Larry’s batting average is .260. If he’s at bat four times, what is the probability that he gets exactly two hits?

Solution:

n = 4, p = 0.260, x = 2

Note: Some textbooks use r for number of successes, rather than x.

Here you want the probability of exactly two successes, so FROM and TO will both be 2. In other words, you are computing the probability of 2 through 2 successes. Run the MATH200D program, select 5:Binomial prob, and specify

n = 4, p = .26, from = 2, to = 2.

The input and output screens are shown at right.

Answer: P(2) = 0.2221

Caution:  Sometimes the probability is very small and the calculator reports it in scientific notation, such as 5.4189E-6. The exponent is not a decoration, and you must not report the probability as simply 5.1489. Probabilities are never greater than 1!

Conventionally, we round probabilities to four decimal places. If the probability is smaller than 0.0001 (smaller than 1E-4), either show it to two significant figures, such as 1.3×10^-6, or report it as <0.0001.

Example 2: Larry’s batting average is .260. If he’s at bat four times, what is the probability that he strikes out all four times?

Solution: Four out of four strikeouts means no hits, so you’ll put these values into the program:

n = 4, p = 0.260, from = 0, to = 0

The output screen is shown at right.

Answer: P(0) = 0.2999

Binomial Probability for a Range of x Values

Example 3: Suppose 65% of the registered voters in Dryden are Republicans. In a random sample of ten registered voters, what’s the probability of fewer than six Republicans?

Solution: “Fewer than six” is zero through five.

n = 10, p = 0.65, 0 ≤ x ≤ 5

Run the MATH200D program, select 5:Binomial prob, and enter

n = 10, p = .65, from = 0, to = 5

Answer: 0.2485

There’s about one chance in four of getting fewer than six Republicans in a random sample of ten registered voters.

6. Histogram of a Binomial Distribution

The previous section showed how to compute the binomial probability of a particular number of successes, or of a range of numbers of successes. But to get an overview of a distribution, a histogram is most helpful. This part of the program will create one for you, and also calculate the mean and standard deviation of the distribution.

Run the MATH200D program and select 6:Binom PD histo.

Example: Suppose 65% of registered voters in Dryden are Republicans. (a) Plot the binomial probability histogram for the random variable “number of Republicans in a random sample of ten registered voters“. (b) Interpret the mean and standard deviation of the distribution.

Solution: n = 10, p = 0.65. Run the program and enter those values when prompted. The program responds by drawing the histogram and displaying μ and σ.

Interpreting the histogram: The bars are each one unit wide, running from 0 to n (0 to 10, for this example), with the dots marking each integer and the y axis marking the minimum possible value of successes, 0. You can see that the most likely result for a sample of ten registered voters is seven Republicans, with six being just slightly less likely. Eight and five are next most likely, then nine and four. A sample of ten is quite unlikely to contain ten or three Republicans, and fewer than three Republicans are extremely unlikely.

Interpreting the mean: The mean of the distribution is 6.5, meaning that if you took a whole lot of random samples of ten registered voters (with replacement), on average a sample would contain 6.5 Republicans.

Interpreting the standard deviation: The standard deviation is about 1.5. While this distribution is not a normal distribution (bell curve), it’s not extremely different from one, and therefore you can say that the Empirical Rule (68–95–99.7 Rule) is not too far wrong.

2σ = about 3, so you would expect roughly 95% of samples to contain 6.5±3 Republicans. 6.5±3 is 3.5 to 9.5, but the integers within that range are 4 to 9, so the standard deviation tell you that roughly 95% of samples of ten registered voters would have four to nine Republicans. (An equivalent statement is that there’s about a 95% chance that a sample of ten voters will contain four to nine Republicans.) If you use the 5:Binomial prob part of the program to compute the actual probability, you find that P(4 ≤ x ≤9) = 0.9605.

optional extra: Tracing the Histogram

As with other histograms, you can press the [TRACE] key and use the [◄] and [►] keys to display the probability of each bar.

The two screen shots show values from the histogram.

The first picture shows that P(4) is 0.1536: there’s a 15.36% probability that a random sample of ten registered voters will contain four Republicans.

The second picture shows that P(0) is “2.7585E”. Unfortunately, the trace doesn’t show the negative exponent, so you don’t know whether the probability is 2.7585E-4 or 2.7585E-94; but you do know that it’s small, less than 0.001. If it’s important to know the exact value, use the 5:Binomial prob part of the program.

7. Normality Check

Example: Consider these vehicle weights (in pounds):

2500, 3250, 4000, 3500, 2900, 4500, 3800, 3000, 5000, 2200

Construct a plot to decide whether these vehicle weights seem to be normally distributed.

Solution: Put the data in any statistics list, then press [PROG], scroll down to MATH200D, and press [ENTER] twice. Select 7:Normality chk.

The program will make the plot and display the sample size n and correlation coefficient r, as shown at right. If the points are clearly linear (close to a straight line), you know that the sample came from a normal distribution; if they are clearly not near a straight line, you know that the sample is non-normal.

Sometimes it’s hard to decide whether the points are close enough to a straight line. For those cases, the program gives you the correlation coefficient r. r is a measure of how close the points lie to a straight line, with 1 being perfectly linear and 0 being completely non-linear. A good rule of thumb is that r ≥ 0.9 usually means the plot is linear and the original points are normally distributed. But always look first at the plot, and follow your eyes: r is there only for the doubtful cases.

For this example, the program computed a correlation coefficient of 0.9936. If it weren’t already obvious from the plot, this would tell us that the original data are approximately normal.

Reference: Variable Usage

Most parts of the MATH200D program leave useful results in variables, which you can use for further calculation. Use the [ALPHA] key. For instance, if you want a value from variable V, press [ALPHA 6 makes V]. Also, if you’re using any variables or statistics lists yourself, you don’t want to be surprised when the program changes their values. Below you’ll find complete information.

If you want to delete the lists to free up memory, press [2nd + makes MEM] [2] [4], scroll down to find each one, and press [DEL].

If you want to delete the single-letter variables, though it’s hardly worth the effort, press [2nd + makes MEM] [2] [2], cursor to each one, and press [DEL].

1:Histograms etc

• A = upper bound of highest class
• M = lower bound of lowest class
• N = number of classes or number of data points
• W = class width
• LD = class marks
• LF = class frequencies
• used by program, not useful afterward: I, X, θ

2:Box-whisker

• LA = second data list, if required
• LB = second frequency list, if required
• LC = third data list, if required
• LD = first data list
• LE = third frequency list, if required
• LF = first frequency list, if required
• used by program, not useful afterward: θ

3:Time series

• N = number of data points
• LD = the numbers from 1 to N
• LF = data list
• used by program, not useful afterward: X

4:Skew/kurtosis

• E = standard error of skewness, equals 0 for a probability distribution
• F = standard error of kurtosis, equals 0 for a probability distribution
• K = kurtosis (to find excess kurtosis, subtract 3)
• M = mean
• N = sample size (number of data points, or total of frequencies for a grouped frequency distribution), equals 1 for a probability distribution
• S = skewness
• X = fourth moment
• V = variance (for standard deviation, take the square root)
• LD = data list or class marks
• LF = class frequencies
• used by program, not useful afterward: θ, GDB0

5:Binomial prob

• N = number of trials
• P = probability of success on one trial
• X = number of successes “from”
• Y = number of successes “to”
• Z = probability (answer)

6:Binom PD histo

• N = number of trials
• P = probability of success on one trial

7:Normality chk

• N = sample size (number of data points)
• LD = data list
• used by program, not useful afterward: X, LF

What’s New

14 Nov 2009: Add note about future replacement of this program

18 Oct 2009: several small edits for clarity and to correct typos

16 Sep 2009: Note the equivalence of class marks and class midpoints, in several places.

24 Aug 2009: Protect the program to prevent accidental editing.

7–8 Mar 2009: Improve the TI-83/84 program parts 1:Histograms etc and 4:Skew/kurtosis, as follows:

• Replace separate “Data list” and “Class marks” prompts with “Class marks or data list” and ask data arrangement after data list, for consistency with 3:Time series.
• Replace generic “Freq/prob dist” data arrangement with “Grouped dist”, “Ungrouped dist”, and “prob dist”. Probability distributions are checked for total probability of 1.
• Allow as few as two classes (previously three).
• Allow more than 47 classes in 4:Skew/kurtosis, and don’t require equal widths except in the grouped frequency distribution.

Update screen shots and text in Histograms and Frequency Polygons and Skewness and Kurtosis accordingly. Create an example for Relative Frequencies or Probabilities, which previously had none. Finally, perform an overdue spell check and correct several typos.

8 Feb 2009: Add a reference to the TI-83 troubleshooting page.

26 Dec 2008: Add a caution about negative exponents in binomial probabilities, and a note on rounding them. Each major section now starts on a new printed page.

13 Dec 2008: Extend the 4:Skew/kurtosis part of the program to allow for probability distributions, where the standard error is zero; and check for fractional frequencies other than in a probability distribution. Add an example of a probability distribution under Skewness and Kurtosis.

7 Dec 2008: Add a program version that works with the original TI-83.

6 Dec 2008:

• pull four separate programs into a unified MATH200D application, and nine Web pages into this one, eliminating a fair amount of duplication; document variable usage clearly
• add 2:Box-whisker, 3:Time series, and 6:Binom PD histo
• change MATH200D program prompts and outputs to lower case or mixed case for readability

Here’s part of the earlier history of the pages now in this page and the programs now part of the MATH200D program:

1. The 1:Histograms etc part of the MATH200D program was created as HISTNPGN on 20 Jan 2008 and documented in Frequency Polygons on TI-83/84. On 7 Jun 2008 the program was moved to a new document, Frequency Histograms and Polygons the Easy Way on TI-83/84; on 21 Sep 2008 the program was rewritten to accommodate lists of numbers as well as frequency distributions and was renamed HISTNPG2. When the program was consolidated into the MATH200D program, I tightened up the error checking and no longer allowed a frequency polygon for a simple list of numbers.
2. The 2:Box-whisker part of the MATH200D program was created on 4 Dec 2008 by following the instructions in Sample Statistics on TI-83/84. Then I added the ability to do plain lists or grouped frequency distributions, and to plot up to three boxplots on a screen.
3. Time Series Plots started life on 15 Sep 2007 as Plotting Time Series on TI-83/84, a set of directions rather than a program. The 3:Time series part of the MATH200D program was created on 3 Dec 2008, and the description was modified to emphasize setting the vertical scale properly and to show how to trace the graph.
4. Skewness and Kurtosis began on 12 Jan 2008 as Skewness and Kurtosis on TI-83/84, with the SKURT program.
5. Probability of a Binomial Experiment began on 11 Mar 2007 as TI-83/84 Shortcuts for Binomial and Discrete PD; it went through a number of subsequent major revisions. On 17 Jun 2008 the BINOMPRB program was created, and that program has now become the 5:Binomial prob selection in the MATH200D program.
6. Histogram of a Binomial Distribution and the program part 6:Binom PD histo were written for this Web page on 6 Dec 2008.
7. Normality Check began on 16 June 2002 as Normality Check; the 7:Normality chk part of the MATH200D program started as the TI-83 program NORMCHEK in October 2007.