Statistics

Binomial probabilities on the TI 83 or 84 calculator

In this article, we will learn how to find binomial probabilities using your TI 83 or 84 calculator. We’re going to assume that you already know how to determine whether or not a probability experiment is binomial and instead just focus on how to use the calculator itself.

There are two functions you will need to use, and each is for a different type of problem. We will look at each of them below, using an example.

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binomialpdf

The binompdf function on your calculator is for finding the probability of exactly some number of successes.

Example

A survey determines that 62% of the attendees at a conference have attended a similar conference in the last year. Suppose that 9 attendees are randomly selected. Find the probability that exactly 4 have attended a similar conference in the last year.

Let X represent the number of attendees that have attended a similar conference in the last year. We are then trying to find:

P(X = 4)

Step 1: Go to the distributions menu on the calculator and select binompdf.

To get to this menu, press:

followed by

Scroll down to binompdf near the bottom of the list.

Press enter to bring up the next menu.

TI83	TI84

Step 2: Enter the required data.

In this problem, there are 9 people selected (n = number of trials = 9). The probability of success is 0.62 and we are finding P(X = 4). How you enter this looks different in each calculator.

TI83

TI84

Type in 9, 0.62, 4) and then press enter. It will always be in this order: binompdf(n, p, c).

ti83-binompdf

ti83-binompdf-example

Fill in the needed information, highlight paste, and then press enter. You will then need to press enter again to get the final answer.

ti84-binompdf-example1

ti84-binompdf-example2

This shows that the probability of exactly 4 successes is about 0.1475. So, the probability that out of the 9 people we selected, exactly 4 have attended a similar conference in the last year is 0.1475.

binomialcdf

The “cdf” in this stands for cumulative. This function will take whatever value we type in, and find the cumulative probability for that value and all the values below it. In other words, this function allows us to calculate the probability of “c or fewer” successes, for some number c.

Example

A survey determines that 62% of the attendees at a conference have attended a similar conference in the last year. Suppose that 9 attendees are randomly selected. Find the probability that 6 or fewer of these attendees have attended a similar conference in the last year.

This is the same example we used before, but now we are finding a different probability. So, we will once again let X represent the number of attendees that have attended a similar conference in the last year. We are now trying to find:

P(X ≤ 6)

This is the type of probability that the binomcdf function is built for!

Step 1: Go to the distributions menu on the calculator and select binomcdf.

To get to this menu, press:

followed by

Scroll down to binomcdf near the bottom of the list.

Press enter to bring up the next menu.

TI83	TI84

Step 2: Enter the required data.

In this problem, there are 9 people selected (n = number of trials = 9). The probability of success is 0.62 and we are finding P(X ≤ 6). How you enter this looks different in each calculator.

TI83

TI84

Type in 9, 0.62, 6) and then press enter. It will always be in this order: binomcdf(n, p, c).

ti83-binomcdf

ti83-binomcdf-example

Fill in the needed information, highlight paste, and then press enter. You will then need to press enter again to get the final answer.

ti84-binomcdf-example1

ti84-binomcdf-example2

This shows that the probability of 6 or fewer successes is about 0.7287. This means that out of the 9 people we randomly selected, the probability that 6 or fewer have attended a similar conference in the last year is 0.7287.

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Other types of probabilities

As with any probability question, there are definitely other questions that could be asked. For example, we could be asked to find the probability of more than 3 successes. This doesn’t seem to exactly fit the pdf or cdf, however, we can still use these functions to find these probabilities as well. You can see how to approach these types of questions here: http://www.mathbootcamps.com/binomial-probabilities-examples/.

Counting with the multiplication rule

Counting is a really tough area of mathematics, but is also really important for understanding real life applications and, later, for finding probabilities. In this article, we will study one particular method used in counting: the multiplication rule.

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The multiplication rule

Imagine you are trying to guess someone’s password. If you know that the password is made up of 5 letters or numbers, we can imagine that you simply guess all 5 at once. But, we could also imagine that you guess the first letter or number, then the second, then the third, and so on. In the end, the result is the same but the way we think about it is different.

If you can think of a process in steps like this, then we can apply the multiplication rule. In order to find the total number of possible passwords, we first think of how many possibilities there are for the first character, how many for the second, and so on. We then multiply these to find the total number of possibilities.

Example – Passwords

A password is 5 characters long and is made up of letters and numbers. How many different passwords like this are possible?

We will think about the number of possibilities for each character individually.

We know the password is made up of letters and numbers. Some observations about this:

There are 26 letters in the alphabet.
There are 10 possible numbers for any for any character: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
In a password, uppercase and lowercase letters are considered different, so there are 26 + 26 = 52 possible letters for any character.

For the first character, this means there are a total of 10 + 52 = 62 possibilities.

For the next character, we still have 62 possibilities. There was no indication that the password can’t use the same letter or number twice. In fact, there are the same number of possibilities for each character. Filling this in and applying the multiplication rule we have:

Example – passwords revisited

A password is 5 characters long, is made up of letters and numbers, and has no repeated characters. How many different passwords like this are possible?

This question is almost identical to the first one, except for one part: no repeated characters. Pay attention to wording like this!!

This tells you that after you have used one number or letter, you can’t use it again. So the number of choices for the first character is still 62. But now, you have used 1 of the characters, so there are only 61 to choose from for the next. This pattern continues after each choice is made.

Example – buying sandwiches

A deli let’s customers build their own sandwich. They can select from three types of bread (wheat, white, and rye), two types of cheese (american or provolone), four types of meat (turkey, roast beef, ham, and salami) and a sauce (regular mayo, hot mayo, bbq). Using these ingredients, how many different sandwiches are possible?

Applications

Really, you could think of any counting problem as a true application problem – if you have ever seen a commercial that says “over 15,000 different varieties” of something like that, then you have seen counting in use!

When you think about counting, there are some other interesting problems that come up. For example, how many 4 digit bank pin codes are possible? (answer: 10,000 – can you use the multiplication rule to figure out why?) Using another rule called the Pigeonhole Principle, we can say that this means that in any group of more than 10,000 people, at least two MUST share the same pin number. But, this assumes people pick codes randomly – so how true is that? I explore that on my personal blog in the following article: pin codes and the birthday problem.

Probability Terminology

Before you can really begin to understand probability questions, there are a few words/phrases that you should be comfortable with. Let’s take a closer look!

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Probability Experiment

When working with probability, we call anything that we can get a result from a probability experiment. You could consider rolling a die, flipping a coin, or randomly choosing a number a probability experiment. Depending on your perspective, you could even call driving to work a probability experiment with the possible results being you arrive on time or you don’t.

Outcome

A single result of a probability is called an outcome (some books use the word “simple event”). When flipping a coin, the possible results or outcomes are heads or tails. In the example about driving to work, the outcomes are “on time” and “not on time”. As you can see, in some cases the perspective matters. Consider another example: rolling two six sided dice. We could look at the sum of the faces (if two sixes com up, then the outcome is 12) or just the faces themselves (6,6). Both of these could be considered outcomes.

Sample Space

The sample space for a probability experiment is the set of all possible outcomes. This is usually written with set notation (curly brackets). For example, going back to a regular 6-sided die the sample space would be:

$S=\{1,2,3,4,5,6\}$

The concept of a sample space is very important and so I’ve actually put together an article with more examples here: What is a sample space?

Event

An event is any group of outcomes from the sample space. If I was flipping two coins, one event is that I get tails at least once. This happens with the outcomes (heads,tails), (tails, heads), and (tails, tails). Events can also be written using set notation. Looking at the event we just talked about, the event of “tails at least once” could be called E and written as $E=\{HT, TH, TT\}$ .

Note that simple events involve only one outcome, for example rolling a 2 on a die. Using set notation: $E = \{2\}$ .

Now when you study a probability question, you can start to apply this terminology to better understand the question being asked and the situation overall. In many cases, finding the probability of an event only involves finding the number of outcomes in the event and the number of outcomes in the sample space (though sometimes that is much harder to do than you would think!).

Reading scatterplots

Scatterplots are used to understand the relationship or association between two variables. Questions like “When the temperature increases, do gas prices also increase?” or “How are changes in the price of gas related to the number of miles people drive each month?” can be answered by studying the pattern in a scatterplot.

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Basic Structure

Given a scatterplot, the variable on the horizontal axis is the predictor (or independent variable) and the variable on the vertical axis is the response (or dependent variable). Using this terminology, a scatterplot is used to understand how the response responds to changes in the predictor. Each point represents the value of the response for a given value of the predictor. These are called observed values.
scatterplot-structure

Patterns

In general, you can categorize the pattern in a scatterplot as either linear or nonlinear. Scatterplots with a linear pattern have points that seem to generally fall along a line while nonlinear patterns seem to follow along some curve. Whatever the pattern is, we use this to describe the association between the variables. If there is no clear pattern, then it means there is no clear association or relationship between the variables that we are studying.

scatterplot-patterns

As you can see above, linear patterns can be thought of as either positive or negative. In a positive pattern, as the value of the predictor increases, so does the value of the response. This shows up in the scatterplot as a linear pattern that rises from left to right. In a negative pattern, as the predictor increases, the value of the response decreases. This is seen as a linear pattern that falls from left to right.

Strength

The strength of the relationship or association between two variables is shown by how close the points are to each other. This is true whether the pattern is linear, nonlinear, positive, or negative.

scatterplot-strength

It can be somewhat subjective to compare the strength of one association to another. For scatterplots with linear patterns, the correlation coefficient can be used to better understand this strength.

Example

Caitlyn has started a business selling textbooks and novels online. In order to better predict her costs, she has been collecting data on the number of books in each shipment she has sent and the weight of the shipment. She plotted the data in the scatterplot below.

example-scatterplot

Use this scatterplot to answer the following questions.
(a) What has Caitlyn used as the predictor? What about the response?
(b) Describe the association between the two variables. Is it weak, strong, or neither?
(c) What was the heaviest shipment Caitlyn made?
(d) One of the shipments weight a little less than 7 pounds. How many books were in this shipment?

(a) What has Caitlyn used as the predictor? What about the response?

Remember that when using a scatterplot, the idea is to understand how the response responds to changes in the predictor. The predictor is always plotted along the horizontal axis and the response along the vertical. Here, the predictor is the number of books and the response is the weight of the shipment.

(b) Describe the association between the two variables. Is it weak, strong, or neither?

This is a positive linear pattern. That means that there is a positive association between the number of books in each shipment and the weight. That is, as the number of books increases, the weight of the shipment also increases. The points aren’t that close together, nor are they that far apart so this pattern isn’t really weak or strong.

(c) What was the heaviest shipment Caitlyn made?

Since the weight was the response, we are looking for the point with the largest value on the vertical axis.
example-scatterplot-largest-y
From this picture, you can see that the heaviest shipment was 8 pounds.

(d) One of the shipments weight a little less than 7 pounds. How many books were in this shipment?

example-scatterplot-reading-a-value
There is only one shipment with a weight just below 7 pounds. This shipment contained 5 books.

Now that you have seen how to read a scatterplot, the natural question may be “How do I make a scatterplot using a dataset?”. You can click this link to learn how to get a quick scatterplot using your TI83 or TI84 calculator: Making a scatterplot with your graphing calculator.

Making two way tables

Two way tables, also known as contingency tables, show frequencies (counts) as they relate to two variables. To complete a two way table for a set of data, you need to determine the variables of interest, their possible values, and then finally, the frequencies. We will look at how to do this using an example and then also look at how to answer questions about a data set using a two way table.

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Example of making a two way table

Suppose that a company is doing market research on a new product and have selected a random sample of potential customers to help choose the most effective TV commercial. Out of the 180 people in the sample 65 viewed the first version, 30 viewed the second version, and the remainder viewed the third. Of those who viewed the first version, 25 indicated that they were likely to buy the product while the rest said they were either unsure or unlikely to buy the product. For those viewing the second version, 20 said they were likely to buy the product and for the third 54 said the same.

Step 1: Identify the variables

There are two variables of interest here: the commercial viewed and opinion. What we call these two variables isn’t that important – just determine which two pieces of information are present in the situation. Everything in the description is about the version of the commercial that the people viewed and their opinion on whether or not they will buy the product.

Step 2: Determine the possible values of each variable.

For the two variables, we can identify the following possible values.

Commercial viewed: version 1, version 2, version 3

Opinion: likely to buy the product, unsure or unlikely to buy the product

Step 3: Set up the table.

Pick one variable to be represented by the rows and one to be represented by the columns. It doesn’t matter which! Then, use the possible values of the variables to represent the rows and columns. Finally, be sure to add a total column and row. It isn’t required, but it is helpful.

two way table setup

Step 4: Fill in the frequencies.

Here, we want to use the problem to determine the frequencies. Let’s go piece by piece through the problem and translate the statements.

First, we see that 180 people were sampled. This must be the overall total which goes in the bottom right corner.

two way table with total

Out of the 180 people in the sample 65 viewed the first version, 30 viewed the second version, and the remainder viewed the third.

Alright, let’s take the rest of this information and add it into the table.

Of those who viewed the first version, 25 indicated that they were likely to buy the product while the rest said they were either unsure or unlikely to buy the product. For those viewing the second version, 20 said they were likely to buy the product and for the third 54 said the same.

two-way-table-2

Finally, any missing values can be found using the column totals. For example, we know that 65 people viewed version 1 and 25 were then likely to buy the product. So, it must be that $65 – 25 = 40$ were unsure or unlikely to buy. Continuing this process, we can complete the table.

two-way-table-3

The totals on the right, or row totals were found by adding across: $25 + 20 + 54 = 99$ and $40 + 10 + 31 = 81$. So that’s it! We have made our two way table. Here it is all by itself with none of the crazy arrows and circles.
Two way table summarizing this data:
two-way-table-final
While it took a bit of reading, it is a very nice summary to represent this data and can be used to answer many different types of questions.

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Observations and vocabulary

To double check your work, always make sure that the totals across rows and down columns are correct.

two-way-table-check

The row totals 99 and 81 as well as the column totals 65, 30, and 85 are called marginal frequencies while values inside the table like 25, 20, 54, 40, 10, 31 are called joint frequencies.

two-way-table-joint-marginal-frequencies

As was mentioned above, using a two way table, you can easily answer a wide variety of questions using the joint or marginal frequencies. In those types of problems, it will be very important that you read carefully so you know WHICH frequency to use!

Scatterplots on the TI83 or TI84 graphing calculator

Scatterplots are used to visualize the relationship or association between two variables. For example, can you say in general that studying more will result in higher grades? We could investigate this by collecting data on how long students studied and perhaps their grade on a final exam and then creating a scatterplot. The overall pattern would help us determine what kind of association time spent studying has with final exam grades.

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On the TI83 or 84 series of graphing calculators, getting a scatterplot is pretty easy. Let’s use an example data set to walk through the process.

Example

The table below shows the heights (inches) and weights (pounds) of seven randomly selected players on the Chicago Cubs active roster.

Height, x	76	72	77	73	69	75	71
Weight, y	240	190	215	185	180	210	210

We will use the height as the independent variable (the predictor) and the weight as the dependent variable (the response). Often a textbook will tell you which to treat as x (predictor) and y (response). If not, be sure to check out the article on reading scatterplots to better understand how to choose these variables.

Step 1: Enter the data.

To enter data in your calculator, press [STAT] and then choose 1.Edit.

edit-table-ti-83-ti-84

Now enter the values for x (the predictor) into L1 and the values for y (the response) into L2. Just type the number and press enter to go to the next space.

data-in-L1-TI83-TI84

Press the right arrow to go over to L2. Notice that we keep the data in the SAME ORDER as the original table.

data-in-L1-and-L2-TI83-TI84

Stept 2: Set up the scatterplot in STAT PLOTS.

To get to the statplot menu, press [2nd] and [Y=] (at the top of the calculator).

stat-plots-TI83-TI84

Press [ENTER] or 1:Plot 1… to get into the next menu. Once in this menu, highlight [ON] and press [ENTER] to turn the plot on and then make sure that the little graph that looks like a scatterplot is selected for TYPE. You can do this by highlighting it and pressing [ENTER].

statplot-scatterplot-ti83-ti84

Step 3: View the scatterplot

Once plot 1 is on you can press [ZOOM] at the top of the calculator and choose 9:ZoomStat to see the scatterplot.

zoom-stat-ti83-ti84

After pressing [ENTER] or typing 9, the plot should come up!

scatterplot-ti83-ti84

It doesn’t have any labels or anything, but you can use this to see if perhaps linear regression is appropriate or just to see what kind of pattern is present. Here, it seems that taller players generally weigh more, but since the points are not very close together, the association between height and weight isn’t very strong.

If you want to see which points are which, you can press [TRACE] and use the arrows to jump from point to point.

Troubleshooting

Problem: When you choose Zoomstat you get an error.

Believe it or not, the most common cause of this is having your calculator in your backpack and having it accidentally type all kinds of stuff in the y-list. I see it every semester! To check, press [Y=] and scroll through to make sure that there is nothing next to any of the y’s on the menu.

y=-ti83-ti84

Make sure to scroll through the whole list! There can’t be anything there! Also, the only plot highlighted at the top should be PLOT1. The others should not be highlighted.

Problem: Error: DIM

This is a dimension error. It means that the number of values in L1 is different than the number of values in L2. Go back and find which number you left out!

Problem: There is no L1 on your calculator

This is an easy fix. Check out this article for how to get L1 back.

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How to read a boxplot

Boxplots are a way of summarizing data through visualizing the five number summary which consists of the minimum value, first quartile, median, third quartile, and maximum value of a data set. In the following lesson, we will look at how to use this information and the basic form of a boxplot to answer questions, therefore helping you understand how to read a boxplot.

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The basic form of a boxplot

If a data set has no outliers (unusual values in the data set), a boxplot will be made up of the following values.

boxplot-no-outliers

But, if there ARE outliers, then a boxplot will instead be made up of the following values.

boxplot-with-outliers

As you can see above, outliers (if there are any) will be shown by stars or points off the main plot. If there are no outliers, you simply won’t see those points. So, now that we have addressed that little technical detail, let’s look at an example to see what kinds of questions we can answer using a boxplot.

Answering questions with a boxplot

The boxplot below shows the high temperatures in Anchorage, Alaska in May 2014*.

boxplot-high-temps-anchorage-may-2014

Use this to answer the following questions.

(a) Are there any outliers in this data set?
(b) What was the lowest high temperature observed in May?
(c) Complete the sentence: “About 25% of days in May had high temperatures warmer than about ______ °F.”
(d) What was the median high temperature in May?
**(e) How many days in May did Anchorage see a high temperature of 65?
**(f) On what dates was the high temperature over 70°F?

Before we answer these, notice that this particular boxplot is vertical instead of horizontal. Depending on the software used, you may see either configuration. The basic form is the same for both.

(a) Are there any outliers in this data set?

There are no stars or other points past the main line in the boxplot, so no, there are no outliers in this data set.

(b) What was the lowest high temperature observed in May?

Since there are no outliers, the main line through the boxplot starts at the minimum value and ends at the maximum value. We are looking for the minimum value here.

boxplot-high-temps-anchorage-may-2014-minimum-marked

First, you need to figure out the scale. Since every other line is labelled and it is counting by 5, the in between lines must represent 2.5°. The minimum looks just about 47.5°, so we will estimate it at 48° and as a final answer we can say “The lowest observed temperature in May was about 48°F.”

This is something you should be comfortable with. That is, we won’t always be able to give an exact answer from the graph depending on the scale. Without the actual data set, we will often have to estimate.

(c) Complete the sentence: “About 25% of days in May had high temperatures warmer than about ______ °F.”

You may think that we need to be able to count values in the data set to answer this question, but actually we don’t! This is a question that can be answered using the fact that the boxplot shows the quartiles. When the data set is placed in order from smallest to largest, these divide the data set into quarters.

five-number-summary

From the picture:

First quartile – Q₁ – about 25% of a data set is smaller than the first quartile and about 75% is above.
Third quartile – Q₃ – about 75% of a data set is smaller than the third quartile and about 25% is above.

Now to actually answer the question! “Complete the sentence: “About 25% of days in May had high temperatures warmer than about ______ °F.” The third quartile is what we need to complete this sentence.

boxplot-high-temps-anchorage-may-2014-third-quartile

It looks like the third quartile is about 66°. So we can write: About 25% of days in May had high temperatures warmer than about 66°F.

(d) What was the median high temperature in May?

The median is shown by the line inside the box of the boxplot. This may not always be in the middle – it depends on the shape of the distribution among other things.

boxplot-high-temps-anchorage-may-2014-median

The median for this data set is between 62.5°F and 65°F, and a bit closer to 65°F than not. I would estimate it at 64°F.

The median high temperature in May was about 64°F.

(e) How many days in May did Anchorage see a high temperature of 65?

This question illustrates one weakness of a boxplot; a weakness that is shared with histograms. Information about individual data values isn’t shown. There is no way to answer this question with a boxplot. We would need to see a dotplot or a stemplot (or the data set itself) to be able to answer this question.

(f) On what dates was the high temperature over 70°F?

Another question where it would be interesting to know the answer! Unfortunately, this is another case where some information is “lost” when making a boxplot. There is no way to tell which temperatures are from which dates. To see that, we would need to use a timeplot or simply a table.

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Conclusion

These last two questions show you that some plots, like boxplots and histograms, are designed to give you a big picture idea of a data set. Through this though, you lose some information about individual values. When making a plot of your own data set, you must consider whether this is important or not and select your plot accordingly.

*Source for this data: Weather Underground

**If you skipped down here, maybe you were suspicious of questions (e) and (f). You are right to be! These can’t be answered by a boxplot alone. The details are given in the answers.

Common shapes of distributions

When making or reading a histogram, there are certain common patterns that show up often enough to be given special names. Sometimes you will see this pattern called simply the shape of the histogram or as the shape of the distribution (referring to the data set). While the same shape/pattern can be seen in many plots such as a boxplot or stemplot, it is often easiest to see with a histogram. In the examples below, we will look at each of these shapes and some of their important properties.

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Bell shaped / symmetric

Histograms that are bell shaped/symmetric appear to have one clear center that much of the data clusters around. As you get away from this center, there are fewer and fewer values.
bell-shaped-histogram
In the histogram above, that center is about 10. Notice that the tallest bars are around this value. The height of the bars is the frequency, or number of data values in a class. For values much smaller or larger than 10, there aren’t nearly as many data values.

This shape comes up frequently in every day life. For example weights and heights (when you look at genders individually) often follow this pattern. Most people are within a certain amount of the typical value with few extremes in either direction.

Left skewed

In distributions that are skewed left, most of the data is clustered around a larger value, and as you get to smaller values, there are fewer and fewer seen in the data set. In the picture, there is essentially a tail going out to the left. You can see this in the histogram below where much of the data (the higher frequency) is around 24 or so. As you move to smaller numbers, there is less and less frequency. This means there are fewer and fewer observations.

skewed-left-histogram

An easy to think about example of data which would have a skewed left distribution is scores on an easy test. Most students would do well, and as you get to lower scores, there would be fewer and fewer students with those scores.

Right Skewed

Just like you saw with a left skewed distribution, distributions that are skewed right have a tail – but this time it is off to the right. This means that the data is generally clustered around a small value and as you look for larger and larger values, there are fewer and fewer.

skewed-right-histogram

Looking at the histogram above, we can see most of the data is centered around 7 or so and that there are fewer and fewer larger data values. If test scores were skewed right it would not be a good thing! It would mean most students did poorly while only a few did well!

Bimodal

You can think of a histogram with a bimodal shape as having two peaks. Instead of one clear center where there is are a lot of observations, there are two. Often this means that you are looking at two different groups and should take a closer look to see if you can separate them.

bimodal-histogram

In the example shown above, there is a peak around 42 or so and a peak around 58 or so. It is almost as if two symmetric/bell shaped histograms were shoved together. In real life, you might see this if you look at a data set for heights of people and it included both men and women. There would be a peak around the typical height of a man and a peak around the typical height of a women.

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Uniform

Data that follows a uniform pattern has approximately the same number of values in each group or class (represented by a bar).

uniform-histogram

The histogram above follows a very uniform pattern as every bar is almost exactly the same height. This type of pattern shows up in some types of probability experiments. For example, if you were to take a 6 sided fair die and roll it many times (as in 100+) you would get a pattern that is approximately uniform.

You will find that the shape of a distribution is important in understanding the data set and in choosing the best measure of center, such as the mean or the median, to represent the data. This is why one of the first steps of analyzing a data set is to always plot your data!

How to read a stemplot

Stemplots are sometimes called stem and leaf plots because each number in the original data set has been broken up into two pieces: a stem and a leaf. When we read stemplots, we need to keep this in mind. For any stemplot, you will see the leaves on the right hand side of the bar. Each leaf represents a single value in the data set. Using this idea, let’s try to answer some questions.

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Answering questions with a stemplot

So let’s suppose that someone has been selling their old baseball card collection online card by card. They have kept track of all of their sales so far and plotted them on the stemplot below.

stemplot-sale-values

Using this, we will answer the following questions about the sale prices so far.

(a) How many cards have they sold?
(b) What was the lowest sale price? the highest?
(c) What is the median sale price?
(d) What is the mode?

From the key, we can see that the leafs represent the ones digit. This is important to consider when answering these questions!

(a) How many cards have they sold?

Since each leaf represents a single data value, we can just count the number of leaves in the stemplot.
stemplot-sale-values-counting
Based on this, so far they have sold 24 total cards.

(b) What is the lowest sale price? the highest?

The values on the stemplot are all in order from smallest to largest (if you look from top to bottom).
stemplot-large-and-small-values
The lowest sale price of any card so far is $34 while the highest is $150.

(c) What is the median sale price?

Recall that the median is the middle value, when the values are placed in order from smallest to largest. If there are an even number of values, then it is the average of the middle two. In this data set, there are 24 values. The middle two values would be the 12th and 13th values.
stemplot-sale-values-median
The median sale price is:

$\dfrac{65 + 72}{2} = \$68.50$

Note that you could also find the mean sale price by adding all the values and dividing by 24, but with the outlier of $150, the median is a better measure of center.

(d) What is the mode?

The mode doesn’t get used too much, but it is a good test of whether or not you can read a stemplot. The mode represents the most commonly observed value. If you look in the row for 6, you will see that 65 was observed 5 times. This means that 5 different sales were for $65, and that $65 is the most common price that a card has sold for thus far.

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Conclusion

While this doesn’t cover absolutely every question you could see from a stemplot, it definitely covers all the skills you would need to answer most. Just remember to pay attention to the key and that each leaf represents a data value and you’ll have no trouble at all. As you continue to learn about stemplots, you might find it useful to now study how to make one.

How to make a stemplot

When you compare them to histograms or boxplots, stemplots (or stem and leaf plots) are much more simple and straightforward to not only put together but also to read. This, along with the fact that you don’t lose information about individual data values is one of the benefits of a stemplot. The only real downside to selecting a stemplot to represent your data set is that it can be overly complex for large data sets and look kind of goofy if the range of your data set is small (I’ll explain this after we see how to make one).

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So let’s use the following small, made-up data set to see how to make a stemplot.

6	12	4	14	35	33	35	37	18	42
45	38	34	34	42	51	58	50	68	72

Step 1: Pick your “stems” and “leaves”

Each of the numbers in your data set can be viewed as being made up of two parts: a stem and a leaf. You want to pick it so that the leaf will be one digit (the reason for this will make sense in a minute). For this data set, the tens digit will be the stem and the ones digit will be the leaf.

stem and leaf

For single digit numbers, we will use a zero as a placeholder for the tens digit. The picture below shows how we will think about this for the first data value of 6.

stem and leaf 2

Step 2: Draw a vertical bar with all of the stems on the left

This step sounds much more complicated than it actually is. We want to look at our data set and figure out the smallest and the largest stems. We will count up without skipping any numbers at all. For this data set:

Smallest value: 6 (so the smallest stem is 0)
Largest value: 72 (so the largest stem is 7)

So on the left of our bar we will write out all the whole numbers from 0 to 7.

stems

That’s way easier once you actually see it right? Notice that even though we don’t have any data values in the 20s that I still put a 2 there. You shouldn’t skip any values!

Step 3: Put each leaf next to its stem, in order

For this step, it might be helpful to put your data set in order from smallest to largest. Here is our original data set.

6	12	4	14	35	33	35	37	18	42
45	38	34	34	42	51	58	50	68	72

Here is the same data set in order from smallest to largest.

4	6	12	14	18	33	34	34	35	35
37	38	42	42	45	50	51	58	68	72

You can do this all at once, but just to make sure you see where all the numbers are coming from, let’s do it in a couple of steps. First, let’s put the 4 and 6 on the plot.

stemplot-1

Now you see what I mean by “put the value next to its stem in order”. Now to finish this, we will continue the same process. Since there are no values with a stem of 2, we just skip that. Also, whenever there are repeats, we will just list the leaf part twice. So you don’t have to scroll, here is the finished stemplot along with the data set.

4	6	12	14	18	33	34	34	35	35
37	38	42	42	45	50	51	58	68	72

stemplot-finished

Also, notice that we added a title (though here this is made up data, so it is a boring title!) and a key. Every stemplot should have a key because it could be that the data values are decimals and a stem of 3 and a leaf of 4 represents 3.4 and not 34! Without a key, how would we know?

Finally, as I mentioned, for some data sets a stemplot might not be as useful. You can probably imagine that if there were 1000 data values that this would be tough to read and seem busy. But also, imagine if all of the values had the same stem! What if your data set was 100, 108, 109, 109, 109, 108, 107, 106, 104… well you get the idea…? Then your stemplot wouldn’t be much better than a list of numbers. There are ways around this (breaking up the stem into two or three parts) but in the end, it might be better to use something like a dotplot for a data set like this.