Pigeonhole Principle I: Paths, Penguins, and Points
A recent question involved the Pigeonhole Principle; we’ll start with an older question to introduce the idea, then the new question and a few old ones.
Sample Standard Deviation as an Unbiased Estimator
Last time, we saw a summary of the difference between population and sample standard deviation, along with other differences in formulas. Here we’ll look at two short answers about the reason for that difference, and then an extensive look at why it’s true. (I’ve taken extra time and space to work through details in order …
Sample Standard Deviation as an Unbiased Estimator Read More »
Formulas for Standard Deviation: More Than Just One!
Last time we introduced standard deviation. Here we’ll look into why two formulas (namely, population and sample standard deviation) are different, and why several different formulas for either are equivalent. We’ll also discover how to update the standard deviation when a new value is added. In doing so, we’ll see some different perspectives than we …
Formulas for Standard Deviation: More Than Just One! Read More »
Standard Deviation and Its Rivals
We’ve had a number of questions about “measures of dispersion”, such as standard deviation, which tell us how much data spreads out, as opposed to “measures of central tendency”, which tell us where the middle of the data is (as we discussed in Three Kinds of “Average” and Mean, Median, Mode: Which is Best?). Why …
Arranging Letters in Words, Revisited
A recent question illustrates well the different ways to solve problems in combinatorics. We’ll see an easy way, another easy way, and a … less suitable … way to solve a set of problems.
Is There More Than One Standard Form for an Equation?
A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.
A Cubic Challenge
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …
An Easy But Impossible Probability Problem
I like looking a little deeper into problems; here we’ll find that although the problem is simple if you take it on its own terms, those terms are actually impossible. Does it matter?
Internal and External Division of a Segment
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
Dividing a Segment: Get the Order Right!
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …