Suppose you draw a random sample and measure the heights of the subjects. As you measure heights, you can create a distribution of heights. This type of distribution is useful when you need to know which outcomes are most likely, the spread of potential values, and the likelihood of different results.
This GXWeb-based utility offers tools to explore probability distributions and hypothesis tesing.
Probabilities for continuous distributions (like heights, IQ scores, even class test results - for a large enough group!) are measured over ranges of values rather than single points. A probability indicates the likelihood that a value will fall within an interval. On a probability plot, the entire area under the distribution curve equals 1.
The most well-known continuous distribution is the normal distribution, which is also known as the Gaussian distribution or the bell curve. This symmetric distribution fits a wide variety of phenomena, such as human height and IQ scores. It has two parameters - the mean and the standard deviation. The lognormal distribution is another common continuous distribution which can fit skewed data.
Distribution parameters are values that apply to entire populations. Unfortunately, population parameters are generally unknown because it is usually impossible to measure an entire population. However, you can use random samples to calculate estimates of these parameters.
The Bell-shaped Curve and Other Probability Distributions
The standard normal (z) distribution (use the Integral tool to explore) \[\int_{\frac{1}{2}}^{\frac{3}{4}}(\frac{e^{\frac{(x)^2}{-2}}}{\sqrt(2\cdot \pi)})dx\]
The general normal distribution (use the a and b sliders to explore) \[\int_{\frac{1}{4}}^{\frac{3}{4}}(\frac{e^{\frac{(x-a)^2}{-2\cdot b^2}}}{b \cdot \sqrt(2\cdot \pi)})dx\]
You can also use ndist(x) as a shortcut!
To explore further, enter commands like:
bell(0,0.25), bell(0.5,0.3) or bell(1,0.35) to plot various normal distributions.
bell(100,15,70,130) OR bell(0,0.25,-0.5,0.5) for the probability of scores within 2 standard deviations of the mean.
bell(100,15,124,130) for the probability of scores between 124 and 130.
By the way, a mean of 100 and standard deviation of 15 are commonly used for IQ scores! So if you have an IQ of 124, what percentage of the population lie above you?
You might also like to try this interactive exploration
Raw Scores
Z-Scores
\(Mean = \)
⇒ 0
\(SD = \)
⇒ 1
\(LowerBound = \)
⇒ 1.6
\(UpperBound = \)
⇒ 6.6667
For a skewed data set, try the lognormal distribution (use the a and b sliders to explore) \[\int_{0.5}^{1}(\frac{e^{\frac{(log(x)-a)^2}{-2\cdot b^2}}}{x\cdot b \cdot \sqrt(2\cdot \pi)})dx\]
Suppose you have two samples from a population (for example, before and after samples from a medical treatment or exam results on a common test for two parallel class groups) and you wish to know if there is a significant difference between these - or whether any differences might just result from chance or error. We might use a statistic such as the Student's t-test to decide.
A t score HIGHER than the p-value from the t-Test Significance Table for that level of significance and degrees of freedom indicates that the difference between the two samples IS significant, and we would REJECT the null hypothesis (that there is no significant difference and any differences may be attributed to error or chance). A t score LOWER than the p cutoff value and we must ACCEPT the null hypothesis.
Hint: When entering mathematical expressions in the math boxes below (f(x), g(x) and h(x)), use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.
Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". Learn More?
Continued fractions are awesome. Every real number, rational and irrational, can be represented by a continued fraction - the rational ones are finite, of course, but both finite and infinite offer some wonderful patterns and opportunities to explore!
Use the f(x) MathBox or the x-value box above to enter a value to express as a continued fraction. You can even share your current continued fraction as a QR Code!
For example, try \(37 \over 15\), \(\pi\), \(\phi\) or even \(\frac{225}{157}\).
Use the CAS Evaluate option from the dropdown menu above to solve an expression or equation in the \(f(x)\) MathBox. Click on the following to try:
Solve(x^2=x+1,x)
Solve(x^2=x+1,x),x
Notice the difference between these two results? Adding an extra \(,x\) at the end forces an \(exact\) rather than a \(numeric\) result!
And some more to try (just click on each line):
pFactor(320) (Prime factors)
sqcomp(x^2-4*x+3,x),x (Complete the Square)
nthroot(10,3) (Enter into the MathBox as \ nthroot [space] 3 [space] 10 [space])
Use the Solve option from the dropdown menu above to solve an expression or equation in the \(f(x)\) MathBox, or enter solve(an equation) in \(f(x), g(x)\) or \(h(x)\).
Use the Derivative option from the dropdown menu above to differentiate a function in the \(f(x)\) MathBox, or enter \(d/dx\)(a function) in \(f(x), g(x)\) or \(h(x)\).
Use the Integral option from the dropdown menu above to integrate a function in the \(f(x)\) MathBox, or enter \(int\)(a function) in \(f(x)\).
For definite integrals, perform the integration as described using the dropdown, and enter end values when requested. Try \(\int(x^2-x-1,0,x)\) for example. Use the \(x\) box or slider to vary this.
Display the tangent (or Normal) of \(f(x)\) for the current value of \(x\) using the Tangent/Normal button, or enter tangent (or tngnt) or normal directly into the graph mathBoxes - set the desired x-value first.
QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available.
The default link here is the Geometry Expressions Showcase, but you can use it as an alternative to sending your data via email, web or Google Cloud. You can even use it to send your own messages!
