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### Course: AP®︎/College Statistics > Unit 3

Lesson 4: Effects of linear transformations# How parameters change as data is shifted and scaled

See how transforming a data set by adding, subtracting, multiplying, or dividing a constant affects measures of center and spread.

## Want to join the conversation?

- I don't understand why the 1st and 3rd quartiles are different on the spreadsheet than when I do it by hand.

First I ordered the numbers

2,3,3,5,5,5,6,7,7,8,10,13

I got the same median = 5.5

but for the IQR I got = 3.5

1st quartile = 4

3rd quartile = 7.5

IQR= 3.5

At6:11you can see the results of the spreadsheet

1st quartile = 4.5

3rd quartile = 7.25

IQR = 2.75

Is there a difference in how the spreadsheet computes the 1st and 3rd quartiles?(17 votes)- The Excel function QUARTILE is considered inaccurate. It treats the quartile like a percentile and then use linear interpolation to get the output. It's newer version QUARTILE.EXC is more preferable.

https://superuser.com/questions/343339/excel-quartile-function-doesnt-work

The algorithm of QUARTILE.EXC can be described somewhat like below:

- Calculate i = (n-1)/4

- 1st quartile = [i]th number + {i} * ([i+1]th number - [i]th number).

Where [i] = the integral part of i, {i} = the decimal part of i.

- 2nd (or 3rd) quartile: Multiply i by 2 (or 3), then do the same process.

For example, in the lesson we have a set of data: 2,3,3,5,5,5,6,7,7,8,10,13

i = (12 + 1)/4 = 3.25

3 * i = 9.75

1st quartile = 3 + 0.25 * (5 - 3) = 3.5

3rd quartile = 7 + 0.75 * (8 - 7) = 7.75

IQR = 7.75 - 3.5 = 4.25

* Tested using QUARTILE.EXC in Excel.

An alternative recursive algorithm can be used where the data set is splitted into halves to find the median of each half, which is similar to what Sal taught.

Both algorithms produce values that separate the data set into groups of 25%. The algorithm implemented in QUARTILE doesn't.(22 votes)

- Here how standard deviation is scaling if we scale data? If i add 5 to all data, SD is not increasing but if we multiply its increasing. But Multiplication is repeated addition, its same thing like adding 5 five times, then why it is scaling?(5 votes)
- Because...

Say for example I have 4 and 8. The difference is 4, or in other words they are 4 apart. The ratio between them is 1 to 2 (8 = 2x4).

Now let's say I multiply both of those numbers by 5. They become 20 and 40. The ratio is exactly the same: 1 to 2. However, the difference between them is much bigger (20 now), because multiplying by the same number doesn't mean that you are adding the same thing. 5 x 4 = 4+4+4+4+4, and 5 x 8 = 8+8+8+8+8.

So, if we imagine that 4 is the mean in the original set, and 8 is another data point called X, X is now a lot farther away after scaling it (remember that standard deviation is just the average distance from the mean).

What DOESN'T change (I think) is the number of standard deviations away from the mean that data point x is. Say the standard deviation of the dataset is 4. 8 is 1 SD from 4. If we scale the data by 5, then the SD becomes 20, the mean is now 20, and data point x is now 40. Data point x is still 1 SD away.(18 votes)

- Is there a Khan Academy-like course for learning excel?(2 votes)
- I'm not sure about that but there are lots of courses on this kind of stuff on LinkedIn Learning.(1 vote)

- why sal evaluate mean and the standard deviation on population rather than on samples?

I think it could be more practical to evaluate them on sample(2 votes)- Evaluating mean and standard deviation on the population rather than on samples provides a complete understanding of the entire dataset. While sample statistics can be useful for making inferences about a larger population, calculating parameters on the entire population allows for a more accurate representation of the data without potential sampling biases.(1 vote)

- can you tell us how to convert units of measurement?(2 votes)
- I think the magnitude of the unit of measurement remains the same, because the domain of distribution remains the same after scaling or shifting, it still define on all real numbers

while the unit of the unit of measurement may change depends on the context, say the transformation transfer the sample from Fahrenheit to degree.(1 vote)

- why when we scale the sd it changes even if the multiplication is repeated addition.(1 vote)
- See Dan Oschrin's answer https://www.khanacademy.org/video/how-parameters-change-as-data-is-shifted-and-scaled?qa_expand_key=kaencrypted_2ffb06387131d81e5e4d96284524157d_3802f1d38254cbbdeb8fd48c835a8e510ea45da6c66fd25520052bcd0ebe9dd1b88e1bcfe344cb2db5461b02a3b4e5fc3b744faba2a1504c731b28892f1d4b4fb5f67897a93bc7e1e74cef36052b2546ada893bedc4aa017bcc74164c9d432ce011e6847366770120686184c4c406cb52d2058fce6d114c68cde2afac119a520f175837be629e92e0e9b27c97af40e617588403eb2aa411b6fe098797842a54e844bc963a61157257096560c776ac34131b96baf39d2bcfa7bdc080ce793b1c5(2 votes)

- I was never taught that adding/subtracting is shifting and multipyling/dividing is scaling. what does it mean here? To me both of them are just the number increase and that is it!(1 vote)
- In statistics, "shifting" refers to adding or subtracting a constant value from each data point, which moves the entire dataset up or down along the number line without changing its relative distribution. "Scaling," on the other hand, involves multiplying or dividing each data point by a constant, which changes the spread or dispersion of the data. The distinction between the two operations is important because they have different effects on the statistical properties of the data.(1 vote)

- Why does Sal suddenly skip to a completely different definition of the IQR?

For the standard deviation, he explained which function from the spreadsheetprogram he selected.

But during the calculation of the IQR he just used some function, without explaining why he chose that one, and without explaining why it leads to a different answer?

(Within excel, I can choose between 2 different quartile functions, but each one will lead to an IQR which is far away from the IQR we learned to compute in the previous unit ...)(1 vote)- The choice of function for computing quartiles may indeed lead to different results. Different functions may use distinct algorithms or assumptions about the distribution of the data, resulting in variations in the calculated quartiles. It's essential to understand the specific function being used and ensure it aligns with the method taught in previous units.(1 vote)

- So bacically:
`For`

.**Standard deviation and IQR**, they do not change if you shift(+or-) the data. But scaling(×) it would change.

For**mean and median**, they both change if you shift or scale the data.

In the case when they do change, they are only*changed*by the value it**shifts**or the**scale factor**. For instance, if X_i +5 , its mean will also be increased by 5. If its scaled up by 5, the mean will be 5 times of the original mean, and the Standard deviation or IQR would also be 5 times of it

data=X_i

tl;dr

+- changes mean and median only

× changes mean, median SD, and IQR.(1 vote) - what happens to the variation?(0 votes)
- When the data is shifted (added or subtracted by a constant), the variation, represented by measures like standard deviation and interquartile range, remains the same. However, when the data is scaled (multiplied or divided by a constant), the variation also scales accordingly. For example, if all data points are multiplied by 2, the spread of the data, as measured by the standard deviation or interquartile range, will also be multiplied by 2.(1 vote)

## Video transcript

- [Instructor] So I have some
data here in the spreadsheet you could use Microsoft
Excel or you could use Google Spreadsheet and we're gonna use the spreadsheet to quickly calculate some parameters. Let's say this is a population. Let's say this is, we're looking
at a population of students and we wanna calculate some parameters and this is their ages, and we wanna calculate
some parameters on that. And so first I'm gonna calculate
it using the spreadsheet, and then we're gonna think about how those parameters change
as we do things to the data. If we were to shift the data up or down or if we were to multiply
all the points by some value, what does that do to
the actual parameters? So the first parameter I'm
gonna calculate as the mean. Then I'm gonna calculate
the standard deviation. Then I'm gonna calculate the median, and then I wanna calculate, let's say, the inter quartile range. Inter, I'll call it IQR. So let's do this. Let's first look at the
measures of central tendencies. So the mean, the function
on most spreadsheets is the average function, and then I cold use my mouse
and select all of these, or I could press Shift
with my arrow button and select all those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice for every
data point I have before, I now have five more than that. So this is my new dataset, or as I'm calling Data+5. Let's see what the mean of that is. So the mean of that, notice, is exactly five more, and the same would have been true if I added or subtracted any number. The mean would change by the
amount that I add or subtract. That shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by
the numbers you have. If all the numbers are five more, you're gonna add five. In this case, how many numbers are there? One, two, three, four,
five, fix, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives
and then you're gonna divide by 12, and so it makes sense
that your mean goes up by five let's think about how the
mean changes if you multiply. So if you take your data and if I were to multiply
it times five, what happens? So this equals this times five. So now all the data points
are five times more. Now what happens to my mean? Notice my mean is now five times as much. So the measures of central tendency, if I add or subtract, well I'm gonna add or subtract
the mean by that amount, and if I scale it up by five or if I scaled it down by five, well my mean would scale up
or down by that same amount, and if you numerically looked
at how you calculate a mean, it would make sense that this
is happening mathematically. Let's look at the other typical measure of central tendency,
and that is the median. To see if that has the same properties. So let's calculate the median here. So once again you order these numbers and just find the middle number. Which isn't too hard, but a computer can do it awfully fast. So that's the median for that dataset. What do you think the medians gonna be if you take all of the data plus five? Well the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the
middle is gonna be five more. So this should be 10.5, and yes, it is indeed 10.5, and what would happen if you
multiply everything by five? Well once again, you still
have the same ordering. It should just multiply that by five. Yup, the middle number's now
gonna be five time larger. So both of these measures
of central tendency, if you shift all the data points, or if you scale them up, you're going to similarly
shift or scale up these measures of central tendency. Now let's think about
these measures of spread. See if that's the same with
these measures of spread. So standard deviation. So STDEV. I'm gonna take the population
standard deviation. I'm assuming that this
is my entire population. So let me analyze it. So let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when
I shift everything by five. Actually, pause the video. What do you think is going to happen? This is a measure of spread. So if you shift, I'll
tell you what I think. If I shift everything by the same amount, the mean shifts but the distance
of everything from the mean should not change. So the standard deviation
should not change, I don't think, in this example, and indeed, it does not change. So if we shift the datasets. In this case we shifted it up by five, or if we shifted it down by one. Your measure of spread, in
this case standard deviation should not change, or at least the standard
deviation measure of spread does not change, but if we scale it, well
I think it should change, because you could imagine
a very simple dataset that things that were a
certain amount of distance from the mean are now going
to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiplied this by five. So scaling the dataset will
scale the standard deviation is a similar way. What about inter quartile range? Where essentially we're
taking the third quartile and subtracting from
that the first quartile to figure out kind of the
range of the middle 50%. Let's do that. We can have the quartile
function equals quartile and then we want to look at our data, and we want the third quartile. So that's gonna calculate
the third quartile. Minus quartile, same data set. So now we wanna select it again. So same dataset, but
this is now going to be the first quartile. So this is gonna give us
our inter quartile range. This calculates the third
quartile in that dataset and this calculates the first
quartile in that dataset. And we get 2.75. Now let's think about whether
the inter quartile range should change. And I don't think it will. Because remember, everything shifts, and even though the first
quartile is gonna be five more, but the third quartile is
gonna be five more as well. So the difference shouldn't change. And indeed look, the
distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference
should scale up by five, and we see that right over there. So the big takeaway here. I just use the example
of shifting up by five and scaling up by five, but you could subtract by any number, and you could divide by a number as well. The typical measures of central tendency mean and median, they both shift and scale as you shift and scale the data, but your typical measures of spread, standard deviation and
inter quartile range, they don't change if you shift the data, but they do change and they
scale as you scale the data.