Class 08 Matlab 07 206s11

MATLAB 7- Statistics normpdf, normcdf,and norminv

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Learning objectives • How to use normpdf, normcdf, and norminv in Matlab • How to make simple pdf and cdf plots • How to make confidence intervals with different ranges than 95%

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Commands • Normpdf- command for normal probability distribution function – Syntax: f = normpdf (x,mu,sigma), where f is pdf at point x. • Normcdf- command for cumulative distribution function – Syntax: f= normcdf (x,mu,sigma) • Cdf values can range between 0-1 • Pdf values can have non-negative values

• Norminv: command for normal inverse cumulative distribution function – Syntax: f = norminv(P, mu, sigma)

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Sample Code • Normpdf:

How to use normpdf at the command line with a vector input: >> mu = 3; >> sigma = 2; >> x = [-0.4 0.4 4.0] x = -0.4000

0.4000

4.0000

>> f = normpdf(x,mu,sigma) f = 0.0470

0.0857

0.1760

Quick and dirty plotting: >> >> >> >>

x = -3:0.1:3; f = normpdf(x,0,1); figure plot(x,f)

• x is ed as an array; f will be an array of pdf values 4

Plotting the normal pdf

For the normal distribution, a large, large proportion of the pdf is contained between mu – 3*sigma and mu + 3*sigma. For plotting a normal pdf, you rarely have to plot beyond those bounds. At 1 sigma, the function is still changing. Very little changes between 3 and 4 sigma.

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Sample Code • Normcdf:

How to use normcdf at the command line with vector input: >> mu = 3; >> sigma = 2; >> x = [-.4 .4 4] x = -0.4000

0.4000

4.0000

>> F = normcdf(x,mu,sigma) F = 0.0446

0.0968

0.6915

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Explaining the relationship between the normal pdf and cdf graphically

Since the cdf is defined as the integral of the pdf (and vice versa), the value of the cdf at x is the same as the area under the curve of the pdf taken from minus infinity to x.

This area gives this value

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clear fprintf('Hey, I''m part of a z-table!\n\n')

Using normcdf to make a z-table

% we will add p1 and p2 values to get the z-value, % just like in a real z-table p1 = 0:0.01:0.09; np1 = length(p1); p2 = 0:0.1:1.5; np2 = length(p2);

If we use mu = 0 and sigma = 1 (the values for the standard normal), then normcdf will return the probabilities found on the z-table.

% top line of table – we do some fancy fprintf formatting % don’t worry if you don’t understand it ncol = 10; fmt = [' ', repmat('\t%3.2f', 1, ncol - 1), '\t%3.2f\n']; fprintf(fmt, p1) % now make a double loop to sequentially print out probabilities % corresponding to z-values, using normcdf; this could also be % done vectorally, but it might be more confusing? for i = 1:np2 % print left-most column, which is p2 in our implementation fprintf('%2.1f',p2(i));

Now you can create your own z-table (more fun than watching paint dry!)

% now loop over all the p1 values, add then to the current % p2 value for this row, and calculate using normcdf for j = 1:np1 z = p2(i)+p1(j); fprintf('\t%5.4f', normcdf(z,0,1)) end % move to a new line fprintf('\n') end

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Results from the script on the previous slide >> makeMeAzTable Hey, I'm part of a z-table!

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

0.00 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332

0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345

0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357

0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370

0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382

0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394

0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406

0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418

0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429

0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441

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Sample Code • Norminv:

to use normcdf at the command line with  How vector input: >> p = [0.1 0.3 0.9] p = 0.1000

0.3000

0.9000

>> x = norminv(p,3,2) x = 0.4369

1.9512

5.5631

Note we get the special confidence interval value for >> x = norminv(0.975,0,1) x = 1.9600 Note norminv doesn’t work for values outside [0 1] Trial>> x = norminv(-0.2,0,1) x = NaN

• x is ed as an array; f will be an array of pdf values

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Norminv and Confidence interval alphas

You can use norminv to calculate z-values for different percentage confidence intervals. The value of z = 1.96 you’ve seen before is for a 95% two-sided confidence interval. It’s the value from the z-table for 0.975 of the probability – which means that 0.025 of the probability is in values greater than 1.96. Since the distribution is symmetric, there is also 0.025 of the probability beyond -1.96. 0.025+0.025 = 0.05 chance you end beyond ±1.96 randomly 95% of the probability is between -1.96 and 1.96

Hey, I'm part of a z-table!

1.5 1.6 1.7 1.8 1.9 2.0

0.00 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772

0.01 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778

0.02 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783

0.03 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788

0.04 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793

0.05 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798

0.06 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803

0.07 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808

0.08 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812

0.09 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817

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Norminv and Confidence interval alphas (2)

What if you want an 80% two-sided confidence interval? That means we want to leave 10% beyond the z-value, and a corresponding 10% below the negative of the zvalue. So we want to find a value on the table close to 0.90. Looking at the table, we see that for z=1.28 we are very close to 0.90, and so we can use 1.28 instead of 1.96 to get a 80% confidence interval.

Hey, I'm part of a z-table!

1.0 1.1 1.2 1.3 1.4 1.5

0.00 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332

0.01 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345

0.02 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357

0.03 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370

0.04 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382

0.05 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394

0.06 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406

0.07 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418

0.08 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429

0.09 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441

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Norminv and Confidence interval alphas (3)

Trial>> norminv(0.9,0,1) ans = 1.2816 A more exact z-value for an 80% confidence interval is 1.2816

But we have a computer. And Matlab. We can do better. The z-table gives cumulative probability values for the standard normal distribution (mu = 0, sigma = 1) Given a probability p, norminv(p,0,1) gives the corresponding z-value

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Today’s ICA • In community page under Weeks and Topics > Class 08 Matlab 07. • Four main questions: – Load data, use normplot, calculate mean, std dev, normpdf – Group data and find mean, std dev – Use normcdf – Use norminv

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