Help Please on Data Ranges: want to change "60% to 80%" top "60% to 95%"

Polar

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Help Please on Data Ranges: want to change "60% to 80%" top "60% to 95%"

Hello;

I am in need of some help. This is my first post, so I hope that I am doing this correctly.

I have a data set of percentages. Let's say the range is from 60% to 80%. I would like to expand the range so the range goes from 60% to 95%, and adjust each data point accordingly. I realize that this will cause a shift in Mean and standard deviation. In an ideal world, I would like to figure out 2 mathematical formulas for doing this two different ways.
1) Such that I can pick the upper limit and all of the data points adjust to the new range.
2) Such that I can choose the mean and all of the data points adjust accordingly.

So the result will be:
a) a wider range
b) the upper data points will have increased more than the lower data points
c) the lowest data point stays the same.

If anyone could please help me out, I would appreciate it.

Thanks.

Polar
 
I must admit that I am at a loss. Why do you want to falsify what your data say? If the actual highest point is 80%, why do you want say it is 95%?

I suspect that you actually want to do something sensible. Let's start with what you want to accomplish and determine if it can be done without falsifying the data.
 
I must admit that I am at a loss. Why do you want to falsify what your data say? If the actual highest point is 80%, why do you want say it is 95%?

I suspect that you actually want to do something sensible. Let's start with what you want to accomplish and determine if it can be done without falsifying the data.

How about we feel bad because everyone in the class did poorly. Rather than punish the students for our pedagogical failure, we want to scale the grades so that we can be more generous?

The problem is two-fold:
1) Anchor
2) Scale

1) Anchor -- Add the same value to everyone's score. This will raise the minimum and maximum and mean by the same amount
2) Scale -- Multiply each score's difference from the lowest score by the required ratio using the highest score.

Min: 60, Mean: 70, Max: 80
1) Anchor Add 10 -- Min:60+10=70, Mean: 70+10 = 80, Max: 80+10=90
2) Scale Max = 90, so (90-60)/(80-60) = 30/20 = 1.5
----- Min: 60+(60-60)*1.5 = 60 + 0*1.5 = 60+0 = 60
----- Mean: 60+(70-60)*1.5 = 60 + 10*1.5 = 60+15 = 75
----- Max: 60+(80-60)*1.5 = 60 + 20*1.5 = 60+30 = 90

Using a combination of Anchor and Scale, you should be able to accomplish just about anything you like.

I was bored. Let's see what you can figure out.
 
If you look at the equation for calculating standard deviation, you will see that it does not change due to addition of constant to the data points. Mean shifts by the same constant.

It does change due to multiplication of constant to the data points. Look at the equation and tell us what do you find?
 
Hello;

I am in need of some help. This is my first post, so I hope that I am doing this correctly.

I have a data set of percentages. Let's say the range is from 60% to 80%. I would like to expand the range so the range goes from 60% to 95%, and adjust each data point accordingly. I realize that this will cause a shift in Mean and standard deviation. In an ideal world, I would like to figure out 2 mathematical formulas for doing this two different ways.
1) Such that I can pick the upper limit and all of the data points adjust to the new range.
2) Such that I can choose the mean and all of the data points adjust accordingly.

So the result will be:
a) a wider range
b) the upper data points will have increased more than the lower data points
c) the lowest data point stays the same.

Let's suppose you want to transform the grades linearly (not the only way to do it, but the easiest). In your example, you want f(60) = 60, and f(80) = 95. There are standard ways to find the equation of a line through two points, here (60,60) and (80,95); one way is to find the slope, then determine what y-intercept will yield the desired values.

Once you've done that, you can see what effect it has on the mean, and reverse-engineer the formula by finding the new upper limit that will change the mean to the desired value (if possible).
 
First, you want a linear function, of the form y= ax+ b, so that you won't "warp" part of the interval- every number in the interval is changed in the same way. Second, you want 60 mapped to 60 and 80 mapped to 95.

So you want to find numbers a and b so that 60= a(60)+ b and 95= a(80)+ b. If you subtract the first equation from the second, "b" is deleted and leaves 35= 20a. a= 35/20= 7/4. Putting that back into the first equation, 60= (7/4)(60)+ b= 105+ b so b= 60- 105= -45. So the new value is given by y= (7/4)x- 45.

Check: If x= 60 then y= (7/4)(60)- 45= 105- 45= 60 and if x= 80, y= (7/4)(80)- 45= 140- 45= 95.
 
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