# Estimation

## Comparing Logarithms With Different Bases

Logarithms are not hard to work with when only one base is involved (as in most real-life problems); but they can be challenging when each log has a different base. Here, we’ll look at a few problems in which we have to compare logarithms with different bases, showing various strategies.

## Normal Approximation … or Not?

(A new question of the week) A recent question (from May) about approximating the binomial distribution with the normal distribution led to some (accidental and otherwise) insights about the method.

## How Far Can I See?

We have been looking at questions about the roundness of the earth, starting with the general fact, and then the determination of the size of the earth. A very common question is about how that roundness affects what we can see, sometimes as a challenge (“If I can see this, then how can the earth …

## Making a Sphere from Flat Material

(A new question of the week) A recent question asking how to make a sphere out of flat material called for a look at an old question on the same topic, and some new ideas, including thoughts about approximation. And we actually get to see the physical result of our assistance, which is rare!

## Grouped Data: Open-ended Classes?

(A new question of the week) A recent question raised a different issue about grouped frequency distributions than we have discussed previously: What do you do when the last class is labelled something like “30 or more”? As we’ll see, there is no one right answer!

## Finding the Median of Grouped Data

(An archive question of the week) Last time we looked at a formula for approximating the mode of grouped data, which works well for normal distributions, though I have never seen an actual proof, or a statement of conditions under which it is appropriate. We have also received questions about a much more well-known, and …

## Finding the Mode of Grouped Data

The mode of a list of data values is simply the most common value (or values … if any). When data is grouped (binned) as in a histogram, we normally talk only about the modal class (the class, or group, with the greatest frequency), because we don’t know the individual values. But some sources teach …

## Distances to an Arc: Exact and Approximate Formulas

(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …

## Measuring Water in a Tank

Over the years, we have received a huge number of questions asking about how to find the amount of liquid (water, oil, …) in a tank, usually a horizontal cylindrical tank. The simplest case involves a rather complicated formula; from there, we can reverse the formula (finding the depth for a given volume), or we …

## A Fermi Problem

(An archive problem of the week) A couple weeks ago, in discussing the value of estimates, I included one example of a (very simple) Fermi problem: one in which it is necessary to invent the data as well as the method of solution. Today, I will examine one answer in which we dug deeper into …