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Explained with Three Examples
The Race and the Naughty Puppy

This starts with some raw data ( not a grouped frequency yet) ...

Alex timed 21 people in the sprint race, to the nearest second:

59, 65, 61, 62, 53, 55, 60, 70, 64, 56, 58, 58, 62, 62, 68, 65, 56, 59, 68, 61, 67

To find the Alex adds up all the numbers, then divides by how many numbers:
Mean = 59 + 65 + 61 + 62 + 53 + 55 + 60 + 70 + 64 + 56 + 58 + 58 + 62 + 62 + 68 + 65 + 56 + 59 + 68 + 61 + 6721
Mean = 61.38095...
To find the Alex places the numbers in value order and finds the middle number.

In this case the median is the 11 th number:

53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70

Median = 61

To find the Mode, or modal value, Alex places the numbers in value order then counts how many of each number. The Mode is the number which appears most often (there can be more than one mode):

53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70

62 appears three times, more often than the other values, so Mode = 62

Grouped Frequency Table

Alex then makes a Grouped Frequency Table:

So 2 runners took between 51 and 55 seconds, 7 took between 56 and 60 seconds, etc

Oh No!

Suddenly all the original data gets lost (naughty pup!)

Only the Grouped Frequency Table survived ...

... can we help Alex calculate the Mean, Median and Mode from just that table?

The answer is ... no we can't. Not accurately anyway. But, we can make estimates.

Estimating the Mean from Grouped Data

So all we have left is:

The groups (51-55, 56-60, etc), also called class intervals, are of width 5

The midpoints are in the middle of each class: 53, 58, 63 and 68

We can estimate the Mean by using the midpoints.

So, how does this work?

Think about the 7 runners in the group 56 - 60: all we know is that they ran somewhere between 56 and 60 seconds:

Maybe all seven of them did 56 seconds,

Maybe all seven of them did 60 seconds,

But it is more likely that there is a spread of numbers: some at 56, some at 57, etc

So we take an average and assume that all seven of them took 58 seconds.

Let's now make the table using midpoints:

Our thinking is: "2 people took 53 sec, 7 people took 58 sec, 8 people took 63 sec and 4 took 68 sec". In other words we imagine the data looks like this:

53, 53, 58, 58, 58, 58, 58, 58, 58, 63, 63, 63, 63, 63, 63, 63, 63, 68, 68, 68, 68

Then we add them all up and divide by 21. The quick way to do it is to multiply each midpoint by each frequency:

And then our estimate of the mean time to complete the race is:

Very close to the exact answer we got earlier.

Estimating the Median from Grouped Data

Let's look at our data again:

The median is the middle value, which in our case is the 11 th one, which is in the 61 - 65 group:

We can say "the median group is 61 - 65"

But if we want an estimated Median value we need to look more closely at the 61 - 65 group.

We call it "61 - 65", but it really includes values from 60.5 up to (but not including) 65.5.

Why? Well, the values are in whole seconds, so a real time of 60.5 is measured as 61. Likewise 65.4 is measured as 65.

At 60.5 we already have 9 runners, and by the next boundary at 65.5 we have 17 runners. By drawing a straight line in between we can pick out where the median frequency of n/2 runners is:

And this handy formula does the calculation:

where:

For our example:

Estimating the Mode from Grouped Data

Again, looking at our data:

We can easily find the modal group (the group with the highest frequency), which is 61 - 65

We can say "the modal group is 61 - 65"

But the actual Mode may not even be in that group! Or there may be more than one mode. Without the raw data we don't really know.

But, we can estimate the Mode using the following formula:

where:

In this example:
Our final result is:
(Compare that with the true Mean, Median and Mode of 61.38..., 61 and 62 that we got at the very start.)

And that is how it is done.

Now let us look at two more examples, and get some more practice along the way!

Baby Carrots Example

Example: You grew fifty baby carrots using special soil. You dig them up and measure their lengths (to the nearest mm) and group the results:

Mean

Median

The Median is the mean of the 25 th and the 26 th length, so is in the 170 - 174 group:

Mode

The Modal group is the one with the highest frequency, which is 175 - 179:

Age Example

Age is a special case.

When we say "Sarah is 17" she stays "17" up until her eighteenth birthday.
She might be 17 years and 364 days old and still be called "17".

This changes the midpoints and class boundaries.

Example: The ages of the 112 people who live on a tropical island are grouped as follows:

A child in the first group 0 - 9 could be almost 10 years old. So the midpoint for this group is 5 not 4.5

The midpoints are 5, 15, 25, 35, 45, 55, 65, 75 and 85

Similarly, in the calculations of Median and Mode, we will use the class boundaries 0, 10, 20 etc

Mean

Median

The Median is the mean of the ages of the 56 th and the 57 th people, so is in the 20 - 29 group:

Mode

The Modal group is the one with the highest frequency, which is 20 - 29:

Summary

For grouped data, we cannot find the exact Mean, Median and Mode, we can only give estimates.

To estimate the Mean use the midpoints of the class intervals:
Estimated Mean = Sum of (Midpoint × Frequency)Sum of Frequency

To estimate the Median use:
where:

L is the lower class boundary of the group containing the median

n is the total number of data

B is the cumulative frequency of the groups before the median group

G is the frequency of the median group

w is the group width

To estimate the Mode use:
where:

L is the lower class boundary of the modal group

fm-1 is the frequency of the group before the modal group

fm is the frequency of the modal group

fm+1 is the frequency of the group after the modal group

w is the group width

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Mean, Median and Mode from Grouped Frequencies

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Learn how to calculate the Mean, Median and Mode from grouped frequencies.

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Video: Link

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Box Plots, Histograms, Dot Plots and Bar Graphs: What they are and when to use

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Box Plots, Histograms, Dot Plots and Bar Graphs: What they are and when to use

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You can look at different types of graphs in terms of how much of the data they actually show. The dot plot shows all of it: every value appears individually. Our age histogram groups the data into 19 classes, so it distills 5000 numbers down to only 19.

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This example shows how to make a histogram. Remember that the horizontal axis represents the values of the variables. The vertical axis gives us the frequency of those variables. For more videos visit http://www.mysecretmathtutor.com

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Frequency tables to help you make a histogram

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Frequency tables to help you make a histogram

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A box and whisker plot is a diagram that shows the statistical distribution of a set of data. This makes it easy to see how data is distributed along a number line, and it's easy to make one yourself! Gather your data. Let's say we start...

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7th Grade IXL- Interpreting Box and Whisker Plots

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7th Grade IXL- Interpreting Box and Whisker Plots

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BB.15 Interpret box-and-whisker plots

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Learn seventh grade math skills for free! Choose from hundreds of topics including proportions, rational numbers, algebra, probability, and more. Start now!

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BB.8 Interpret histograms

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Learn seventh grade math skills for free! Choose from hundreds of topics including proportions, rational numbers, algebra, probability, and more. Start now!

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Given a random sample, practice figuring out what can we reasonably infer about the entire population?

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Resource for Statistics: Definitions and activities

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Resource for Statistics: Definitions and activities

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Math is Fun website

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How far, on average, all values are from the middle.

Calculating It

Find the mean of all values ... use it to work out distances ... then find the mean of those distances!

In three steps:

1. Find the mean of all values

2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)

3. Then find the mean of those distances

Like this:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Step 2: Find the distance of each value from that mean:

Which looks like this:
(No minus signs!)
Step 3. Find the mean of those distances:

So, the mean = 9, and the mean deviation = 3.75

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For deviation just think distance

Formula

The formula is:

Σ is Sigma, which means to sum up

|| (the vertical bars) mean Absolute Value, basically to ignore minus signs

x is each value (such as 3 or 16)

μ is the mean (in our example μ = 9)

N is the number of values (in our example N = 8)

Let's look at those in more detail:

Absolute Deviation

Each distance we calculate is called an Absolute Deviation, because it is the Absolute Value of the deviation (how far from the mean).

|-3| = 3

For any value x:

Absolute Deviation = |x - μ|

From our example, the value 16 has Absolute Deviation = |x - μ| = |16 - 9| = |7| = 7

And now let's add them all up ...

Sigma

The symbol for "Sum Up" is Σ (called Sigma Notation), so we have:

Sum of Absolute Deviations = Σ|x - μ|

Divide by how many values N and we have:

Let's do our example again, using the proper symbols:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Step 2: Find the Absolute Deviations:

Step 3. Find the Mean Deviation:
Note: the mean deviation is sometimes called the Mean Absolute Deviation (MAD) because it is the mean of the absolute deviations.
What Does It "Mean" ?

Mean Deviation tells us how far, on average, all values are from the middle.

Here is an example (using the same data as on the Standard Deviation page):

Example: You and your friends have just measured the heights of your dogs (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Step 1: Find the mean:

Step 2: Find the Absolute Deviations:

Step 3. Find the Mean Deviation:

So, on average, the dogs' heights are 127.2 mm from the mean.

(Compare that with the Standard Deviation of 147 mm)

A Useful Check

The deviations on one side of the mean should equal the deviations on the other side.

From our first example:

Example: 3, 6, 6, 7, 8, 11, 15, 16

The deviations are:

Likewise:

Example: Dogs

Deviations left of mean: 224 + 94 = 318

Deviations right of mean: 206 + 76 + 36 = 318

If they are not equal ... you may have made a msitake!

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How far, on average, all values are from the middle. Find the mean of all values ... use it to work out distances ... then find the mean of those distances! In three steps: 1. Find the mean of all values 2.

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Biased and Unbiased Flashcards Vocabulary

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Biased and Unbiased Flashcards Vocabulary

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Biased and Unbiased Samples Flashcards | Quizlet

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Start studying Biased and Unbiased Samples. Learn vocabulary, terms, and more with flashcards, games, and other study tools.

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Sampling and Populations PPT Try these examples!

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Sampling and Populations PPT Try these examples!

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Extension: How do statistics work in the real world? Check out Gapminder interactives!

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Extension: How do statistics work in the real world? Check out Gapminder interactives!

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Animated global statistics that everyone can understand

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In this lesson you will learn how to collect data about a population by identifying a sample of the population.

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- [Instructor] Let's say that we run a school and in that school there is a population of students right over here. And that is our population. And we want to get a sense of how these students feel about the quality of math instruction at the school, so we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead, we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We are going to sample that population. Now in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random. Not random. And it might seem, at first, pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So, simple, simple, random, random, sample, and this is saying, alright, let me maybe assign a number to every person in the school, maybe they already have a student ID number, and I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students, so let's say there's a sample of 100 students, that I'm going to apply the survey to, so that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population, we are just randomly picking people out, and we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick the students. Now that's pretty good, it's unlikely that you're going to have bias from this sample, but there is some probability that, just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors, and that's a possibility. So even though you are taking a simple random sample that is truly random, once again, it's some probability that it's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we want to, we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get a appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors, and then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors, and instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school, and there is some probability, you know, if I do 100 random people, it's very likely that it's close to 50/50, but there's some chance, just due to randomness, there's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here, clustered, a clustered sample, and what we do is we sample groups. Each of those groups we feel confident has a good balance of male females. So, for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there as you can tell this is not a trivial thing to do, let's just say that we can split, let's say we can split our population into groups, maybe these are classrooms, and each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms, so that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close or maybe a exact balance of males and females so we know that we're gonna get good representation, but we are still sampling, we are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So, once again, these are all forms of random surveys, or random samples, you have the simple random sample, you can stratify, or you can cluster and then randomly pick the clusters and then survey everyone in that cluster. Now if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary survey, or voluntary sample, and this might just be you tell every student at the school, "Hey, here's a web address. "If you're interested, come and fill out this survey." And that's likely to introduce bias because you might have maybe the students who really like the math instruction at their school more likely to fill it out, maybe the students who really don't like it are more likely to fill it out, maybe it's just the kids who have more time more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other because, you know, they volunteered for it. Another not random sample would be called you're introducing bias because of convenience is the term that's often used, and this might say, well, let's just sample the 100 first students who show up in school. And that's just convenient for me because I didn't have to use random numbers, or do the stratification, or doing any of this clustering, but you can understand how this also would introduce bias, because the first 100 students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor where they're all happy about it. Or it might go the other way, the instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer or you just say, "Oh, let me do the first N students." Or you say, "Hey, let me just talk to all of the students "who happen to be in front of me right now." They might be in front of you out of convenience, but they might not be a true random sample. Now there is other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, "Well, yeah, I guess I feel lucky." Well, if the wording was, do you like the fact that a disproportionate more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there is a lot that would go into this. And the other one is just people's, you know, it's called response bias. And, once again, this isn't about... Response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful, and so they might be overly positive or not fill it out at all. So anyway, this is a very high level overview of how you could think about sampling. You want to go random because it lowers the probability of their introducing some bias into it. And then these are some techniques. And also think about whether you're falling into some of these pitfalls that have a good chance of introducing bias.

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Techniques for random sampling and avoiding bias (video) | Khan Academy

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Instructor] Let's say that we run a school and in that school there is a population of students right over here. And that is our population. And we want to get a sense of how these students feel about the quality of math instruction at the school, so we construct a survey, and we just need to decide who are we going to get to actually answer this survey.

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We used an SRS sample. WE also used RNG to find which row to add. We randomly selected 10 rows. We found the mean for each, and then we found the mean for each row and then we found the mean of each row, then our answer was 46. Another way to find the mean is to add the lowest and the highest number 57 and 47, and dividing by 2, and the answer was 52.

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I used a simple random sample to choose who to survey by using a number generator. My sample size is 30 people because it is 30 percent of one hundred and it say that it is a good sample size. I got $4.698. This would be use so you did not have to sample everyone.