# bell graph for standard deviation

#### leejohnson222

Joined Jan 11, 2023
23
i need to create 2 graphs to show if changing the standard deviation and mean helps in getting closer to a nominal value

1st one is in a range of 19k -24k mean = 21.5k s.d = 2k for this i get -1.25 and 1.25

2nd one same range but mean is 21.8 and sd = 1k
for this i get 19-21.8 = -2.8 then 24 -21.8 = 2.2
1 1
in my text book im told to have the mean as the centre point and use s.d as each unit on the x axis but the 2 will look a little different as im trying to make them equal across the X axis and stick to their s.d units.
Hope this makes sense, ive been stuck on this all day

#### Papabravo

Joined Feb 24, 2006
19,845
i need to create 2 graphs to show if changing the standard deviation and mean helps in getting closer to a nominal value

1st one is in a range of 19k -24k mean = 21.5k s.d = 2k for this i get -1.25 and 1.25

2nd one same range but mean is 21.8 and sd = 1k
for this i get 19-21.8 = -2.8 then 24 -21.8 = 2.2
1 1
in my text book im told to have the mean as the centre point and use s.d as each unit on the x axis but the 2 will look a little different as im trying to make them equal across the X axis and stick to their s.d units.
Hope this makes sense, ive been stuck on this all day
I'm not clear on what you are calculating or why you are calculating it. There is no a priori reason why the mean of the population distribution should be in the middle of the range of a sample from that population. You are taking the difference between the extremes of the range and converting to units of standard deviation. In this first case the mean is in the center of the range, and in the second case it is not. I don't know what conclusion you think that implies. To me the second case implies a higher probability that a random sample from the population will be within the range, since the range includes all values around the mean within [-2.8σ, 2.2σ]

This gets into deep questions about whether the mean of the sample is an accurate estimate of the mean of the population. Save for the variance or standard deviation which is the square root of the variance. If we have two samples from the population and we get different results for the mean and the standard deviation, what are we supposed to conclude?

#### leejohnson222

Joined Jan 11, 2023
23
ok its probably my terrible explanation
my question is 500 resistors with a mean of 21.5k and a s.d of 2k how many resistors could you expect in a range of 19-24k
with a graph to show this,

then to try and increase better production of the resistors to get more closer to the desired 22k show a second graph using the same range but with a mean of 21.8k and a s.d of 1k.

compare the 2 and say which if the change in figures were a benefit

#### WBahn

Joined Mar 31, 2012
28,172
So draw/plot/sketch a Gaussian curve on a set of axis from, say, 15 kΩ to 30 kΩ. Try to be as accurate as you can by carefully plotting points every, say, half standard deviation. The total area under this curve should be 1.0 (if you go all the way out to infinity in both directions). Now draw vertical lines at 19 kΩ and 24 kΩ. The area under the curve between these limits represents the fraction of resistors that fall within that range.

You can either integrate the Gaussian curve between these limits (not the funnest of times, but not bad it you just do it numerically in a spreadsheet), or you can use the cumulative probability distribution (or the closely related standard error function, erf(), or the Q function). Most spreadsheets support at least the erf() function.

#### leejohnson222

Joined Jan 11, 2023
23
ok great, i need to hand draw these and compare, so i will draw what i believe to be the results for the first set of numbers and then for the second set of numbers. I believe i will then need to write a short explanation on how you can see with the 2 graphs that one should yeald a better number of resistors closer to the desired 22k ohms or a more towards that top range. I can also explain that the standard deviation is lower and the mean is closer to the desired 22k ohms.

#### leejohnson222

Joined Jan 11, 2023
23
so this is the rough version i sketched, i will try to get a better one done, i think this shows that one graph has a much larger number of resistors that will fall within the range, it might be useful to have a second X axis showing the 15 -30oms
as you said because i have used z scores here. I think the 2nd set of numbers should have better results as the stadard deviation is lower and the mean is slightly closer to the desired 22koms

maybe this type of example would be better, showing range and z score on the axis

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#### WBahn

Joined Mar 31, 2012
28,172
Looks like you're on the right track, but keep in mind that the problem asked specifically for how many of the 500 resistors are expected to be within the limits given.

I didn't see that the problem specified that the target was 22 kΩ. It gave limits of 19 kΩ and 24 kΩ, which would indicate that the nominal resistance was 21.35 Ω +/- 11%.

#### leejohnson222

Joined Jan 11, 2023
23
sorry thats my mistake, that is later in the question and it states nomimal desired target is 22koms, yes it does ask for an expected number out of 500 resistors and i came to 394