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A histogram is a graphical display of data using bars of different heights. In a histogram, each bar groups numbers into ranges. Taller bars show that more data falls in that range. A histogram displays the shape and spread of continuous sample data.

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Histograms - Understanding the properties of histograms, what they show, and when and how to use them | Laerd Statistics

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Histograms - Understanding the properties of histograms, what they show, and when and how to use them | Laerd Statistics

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Histogram: a graphical display of data using bars of different heights.

It is similar to a Bar Chart, but a histogram groups numbers into ranges .

The height of each bar shows how many fall into each range.

And you decide what ranges to use!

Example: Height of Orange Trees

You measure the height of every tree in the orchard in centimeters (cm)

The heights vary from 100 cm to 340 cm

You decide to put the results into groups of 50 cm:

So a tree that is 260 cm tall is added to the "250-300" range.

And here is the result:

You can see (for example) that there are 30 trees from 150 cm to just below 200 cm tall

(PS: you can create graphs like that using Make your own Histogram)

Notice that the horizontal axis is continuous like a number line:

Example: How much is that puppy growing?

Each month you measure how much weight your pup has gained and get these results:

0.5, 0.5, 0.3, −0.2, 1.6, 0, 0.1, 0.1, 0.6, 0.4

They vary from −0.2 (the pup lost weight that month) to 1.6

Put in order from lowest to highest weight gain:
−0.2, 0, 0.1, 0.1, 0.3, 0.4, 0.5, 0.5, 0.6, 1.6
You decide to put the results into groups of 0.5:

And here is the result:

(There are no values from 1 to just below 1.5, but we still show the space.)

The range of each bar is also called the Class Interval

In the example above each class interval is 0.5

Histograms are a great way to show results of continuous data, such as:

But when the data is in categories (such as Country or Favorite Movie), we should use a Bar Chart.

Frequency Histogram

A Frequency Histogram is a special graph that uses vertical columns to show frequencies (how many times each score occurs):

Here I have added up how often 1 occurs (2 times),
how often 2 occurs (5 times), etc,
and shown them as a histogram.

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Histogram: a graphical display of data using bars of different heights. It is similar to a Bar Chart, but a histogram groups numbers into ranges . The height of each bar shows how many fall into each range. And you decide what ranges to use!

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Definitions:

The lower half of a data set is the set of all values that are to the left of the median value when the data has been put into increasing order.

The upper half of a data set is the set of all values that are to the right of the median value when the data has been put into increasing order.

The first quartile, denoted by Q , is the median of the lower half of the data set. This means that about 25% of the numbers in the data set lie below Q ₁ and about 75% lie above Q ₁ .

The third quartile, denoted by Q , is the median of the upper half of the data set. This means that about 75% of the numbers in the data set lie below Q ₃ and about 25% lie above Q ₃ .

Example 1: Find the first and third quartiles of the data set {3, 7, 8, 5, 12, 14, 21, 13, 18}.

First, we write data in increasing order: 3, 5, 7, 8, 12, 13, 14, 18, 21.

As on the previous page, the median is 12.

Therefore, the lower half of the data is: {3, 5, 7, 8}.

The first quartile, Q ₁, is the median of {3, 5, 7, 8}.

Since there is an even number of values, we need the mean of the middle two values to find the first quartile:

.

Similarly, the upper half of the data is: {13, 14, 18, 21}, so

.

Example 2: Find the first and third quartiles of the set {3, 7, 8, 5, 12, 14, 21, 15, 18, 14}.

Note that here we consider the two 14's to be distinct elements and not representing the same item; consider this like you obtained a score of 14 on two different quizzes.

First, we write the data in increasing order: 3, 5, 7, 8, 12, 14, 14, 15, 18, 21.

As before, the median is 13 (it is the mean of 12 and 14 - the pair of middle entries).

Therefore, the lower half of the data is: {3, 5, 7, 8, 12}.

Notice that 12 is included in the lower half since it is below the median value.

Then Q (there are five values in the lower half, so the middle value is the median). Similarly, the upper half of the data is: {14, 14, 15, 18, 21}, so Q.

Self Check Problem

The following dollar amounts were the hourly collections from a Salvation Army kettle at a local store one day in December: $19, $26, $25, $37, $32, $28, $22, $23, $29, $34, $39, and $31. Determine the first quartile and third quartile for the amount collected.
Solution
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Definitions: The lower half of a data set is the set of all values that are to the left of the median value when the data has been put into increasing order. The upper half of a data set is the set of all values that are to the right of the median value when the data has been put into increasing order.

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Quartiles are the values that divide a list of numbers into quarters:

Put the list of numbers in order

Then cut the list into four equal parts

The Quartiles are at the "cuts"

Like this:

Example: 5, 7, 4, 4, 6, 2, 8

Put them in order: 2, 4, 4, 5, 6, 7, 8

Cut the list into quarters:

And the result is:

Sometimes a "cut" is between two numbers ... the Quartile is the average of the two numbers.

Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8

The numbers are already in order

Cut the list into quarters:

In this case Quartile 2 is half way between 5 and 6:

Q2 = (5+6)/2 = 5.5

And the result is:

Interquartile Range

The "Interquartile Range" is from Q1 to Q3:

To calculate it just subtract Quartile 1 from Quartile 3, like this:

Example:

The Interquartile Range is:

Q3 − Q1 = 7 − 4 = 3

Box and Whisker Plot

We can show all the important values in a "Box and Whisker Plot", like this:

A final example covering everything:

Example: Box and Whisker Plot and Interquartile Range for

4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11

Put them in order:

3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18

Cut it into quarters:

In this case all the quartiles are between numbers:

Also:

So now we have enough data for the Box and Whisker Plot:

And the Interquartile Range is:

Q3 − Q1 = 15 − 4 = 11

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Quartiles are the values that divide a list of numbers into quarters: Put the list of numbers in order Then cut the list into four equal parts The Quartiles are at the "cuts" Like this: Put them in order: 2, 4, 4, 5, 6, 7, 8 Cut the list into quarters: And the result is: Sometimes a "cut" is between two numbers ...

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Find the mean, median, and mode of the following sets of numbers. And they give us the numbers right over here. So if someone just says the mean, they're really referring to what we typically, in everyday language, call the average. Sometimes it's called the arithmetic mean because you'll learn that there's other ways of actually calculating a mean. But it's really you just sum up all the numbers and you divide by the numbers there are. And so it's one way of measuring the central tendency. The average, I guess, we could say. So this is our mean. We want to average 23 plus 29-- or we're going to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25, and then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then, the next one is 21. There's no 22 here. Let's see, there's two 23's. 23 and a 23. So 23 and a 23. And no 24's. There's a 25. 25. There's no 26, 27, 28. There is a 29. 29. Then you have your 32. 32. And then you have your 33. 33. So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles. If you have an even number, there's actually two numbers that qualify for close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them. That, by itself, can't be the median because there's three less than it and there's four greater than it. And 25 by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle. And I want to be clear, there's no one way of doing it. This is one way of measuring the middle. Let me put that in quotes. The middle. If you had to represent this data with one number. And this is another way of representing the middle. Then finally, we can think about the mode. And the mode is just the number that shows up the most in this data set. And all of these numbers show up once, except we have the 23, it shows up twice. And since because 23 shows up the most, it shows up twice. Every other number shows up once, 23 is our mode.

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Mean, median, & mode example (video) | Khan Academy

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Find the mean, median, and mode of the following sets of numbers. And they give us the numbers right over here. So if someone just says the mean, they're really referring to what we typically, in everyday language, call the average. Sometimes it's called the arithmetic mean because you'll learn that there's other ways of actually calculating a mean.

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A typical example would be the case where nearly every person in a given population lives on about two dollars a day, but there is a small elite with incomes in the millions. The numerical average can mislead by suggesting that the average (in this case, we mean "typical") person earns a few tens of thousands per year. But this does not accurately reflect what we mean when we're trying to discuss the "average" income. This is why the average income is typically expressed by a different sort of average.

What are the three averages?

The three averages are:

The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers.

The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order from smallest to largest, so you may have to rewrite your list before you can find the median.

The "mode" is the value that occurs most often. If no number in the list is repeated, then there is no mode for the list.

The "range" of a list a numbers is just the difference between the largest and smallest values. It expresses "spread", being how far the values are distributed (or how concentrated they are).

Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average, so I'll add and then divide:

Note that the mean, in this case, isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers; you should not be surprised when it isn't.

The median is the middle value, so first I'll have to rewrite the list in numerical order:

There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:

So the median is 14.

The mode is the number that is repeated more often than any other, so 13, I see from my listing above, is the mode.

The largest value in the list is 21, and the smallest is 13, so the range is 21 − 13 = 8.

Note: The formula for the place to find the median is "([the number of data points] + 1) ÷ 2", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer, especially if your list is short. Either way will work.

Find the mean, median, mode, and range for the following list of values:

1, 2, 4, 7

The mean is the usual average:

The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers.

Because of this, the median of the list will be the mean (that is, the usual average) of the middle two values within the list. The middle two numbers are 2 and 4, so:

So the median of this list is 3, a value that isn't in the list at all.

The mode is the number that is repeated most often, but all the numbers in this list appear only once, so there is no mode.

The largest value in the list is 7, the smallest is 1, and their difference is 6, so the range is 6.

The values in the list above were all whole numbers, but the mean of the list was a decimal value. Getting a decimal value for the mean (or for the median, if you have an even number of data points) is perfectly okay; don't round your answers to try to match the format of the other numbers.

Find the mean, median, mode, and range for the following list of values:

8, 9, 10, 10, 10, 11, 11, 11, 12, 13

The mean is the usual average, so I'll add up and then divide:

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5-th value; the formula is reminding me, with that "point-five", that I'll need to average the fifth and sixth numbers to find the median. The fifth and sixth numbers are the last 10 and the first 11, so:

The mode is the number repeated most often. This list has two values that are repeated three times; namely, 10 and 11, each repeated three times.

The largest value is 13 and the smallest is 8, so the range is 13 − 8 = 5.

As you can see, it is possible for two of the averages (the mean and the median, in this case) to have the same value. But this is not usual, and you should not expect it.

Note: Depending on your text or your instructor, the above data set may be viewed as having no mode rather than having two modes, because no single solitary number was repeated more often than any other. I've seen books that go either way on this; there doesn't seem to be a consensus on the "right" definition of "mode" in the above case. So if you're not certain how you should answer the "mode" part of the above example, ask your instructor before the next test.

How can I keep straight which average is which?

About the only hard part of finding the mean, median, and mode is keeping straight which "average" is which. Use this list:

mean: regular meaning of "average"

median: middle value

mode: most often

(In the above, I've used the term "average" rather casually. The technical definition of what we commonly refer to as the "average" is technically called "the arithmetic mean": adding up the values and then dividing by the number of values. Since you're probably more familiar with the concept of "average" than with "measure of central tendency", I used the more comfortable term.)

A student has gotten the following grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall for the course. Assuming all tests are evenly weighted, what is the minimum grade he must get on the last test in order to achieve that overall average?

The minimum grade for the last test is what I need to find. To find the average of all his grades (the known ones, plus the unknown one), I have to add up all the grades, and then divide by the number of grades. Since I don't have a score for the last test yet, I'll use a variable to stand for this unknown value: "". Then, setting the expression for the average equal to the desired average, the computation is:

Multiplying through by 5 and simplifying, I get:

This is the score that he needs on the last test in order to get the overall grade that he is wanting. (If I doubt myself, I can always plug this value into the formula for the average, and confirm that I get 85 as the result.)

You can use the Mathway widget below to practice finding the median. Try the entered exercise, or type in your own exercise. Or try entering any list of numbers, and then selecting the option - mean, median, mode, etc - from what the widget offers you. Then click the button to compare your answer to Mathway's.
(Click here to be taken directly to the Mathway site, if you'd like to check out their software or get further info.)
URL: https://www.purplemath.com/modules/meanmode.htm

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Mean, median, and mode: they're *all* averages..?

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A typical example would be the case where nearly every person in a given population lives on about two dollars a day, but there is a small elite with incomes in the millions. The numerical average can mislead by suggesting that the average (in this case, we mean "typical") person earns a few tens of thousands per year.