average

New Release for Kindle: Math Shorts – Integrals

Yesterday I released the second part of my “Math Shorts” series. This time it’s all about integrals. Integrals are among the most useful and fascinating mathematical concepts ever conceived. The ebook is a practical introduction for all those who don’t want to miss out. In it you’ll find down-to-earth explanations, detailed examples and interesting applications. Check out the sample (see link to product page) a taste of the action.

Important note: to enjoy the book, you need solid prior knowledge in algebra and calculus. This means in particular being able to solve all kinds of equations, finding and interpreting derivatives as well as understanding the notation associated with these topics.

Click the cover to open the product page:

cover

Here’s the TOC:

Section 1: The Big Picture
-Anti-Derivatives
-Integrals
-Applications

Section 2: Basic Anti-Derivatives and Integrals
-Power Functions
-Sums of Functions
-Examples of Definite Integrals
-Exponential Functions
-Trigonometric Functions
-Putting it all Together

Section 3: Applications
-Area – Basics
-Area – Numerical Example
-Area – Parabolic Gate
-Area – To Infinity and Beyond
-Volume – Basics
-Volume – Numerical Example
-Volume – Gabriel’s Horn
-Average Value
-Kinematics

Section 4: Advanced Integration Techniques
-Substitution – Basics
-Substitution – Indefinite Integrals
-Substitution – Definite Integrals
-Integration by Parts – Basics
-Integration by Parts – Indefinite Integrals
-Integration by Parts – Definite Integrals

Section 5: Appendix
-Formulas To Know By Heart
-Greek Alphabet
-Copyright and Disclaimer
-Request to the Reader

Enjoy!

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Analysis: Size and Loading Times of WordPress.com Blogs

In the fast paced online world people are not so patient as in real life. Accordingly, having a large home page size and loading time can negatively affect your blog traffic. Studies have shown that the greater the loading time, the higher the bounce rate. To find out how well my blog performs with respect to this (feel free to use the results for your benefits as well), I did a analysis of 70 WordPress.com blogs. I used iWEBTOOLS’s Website Speed Test and OriginPro for that. With the tool you can analyze ten webpages at once, but note that after ten queries you have to wait a full day (not an hour as the website claims) to do more analysis.

The average size of a WordPress.com blog according to the analysis is 65.3 KB with a standard error SE = 3.0 KB. Here’s how the size is distributed:

WPSize

The average loading time at my internet speed (circa 600 KB/s) is 0.66 s with the standard error SE = 0.10 s. Here’s the corresponding distribution:

WPLoading2

Note that the graph obviously depends on your internet speed. If you have faster internet, the whole distribution will shift to the left. My blog has a home page size of 81.6 KB. From the first graph I can deduce that only about 24 % of home pages are larger in size. My loading time is 0.86 s, here only about 22 % top that. So it looks like I really have to throw off some weight.

Here’s the loading time plotted against the home page size:

WPLoadingSize

In a very rough approximation we have the relation:

loading time = 0.009 * size

In other words: getting rid of 10 KB should lower the loading time by about 0.1 seconds. Now feel free to check your own blog and see where it fits in. If you got the time, post your results (if possible including URL, size, loading time, internet speed) in the comments. I’d greatly appreciate the additional data. For a reliable result regarding loading time it’s best to check the same page three times and do the average.

The Standard Error – What it is and how it’s used

I smoke electronic cigarettes and recently I wanted to find out how much nicotine liquid I consume per day. I noted the used amount on five consecutive days:

3 ml, 3.4 ml, 7.2 ml, 3.7 ml, 4.3 ml

So how much do I use per day? Well, our best guess is to do the average, that is, sum all the amounts and divide by the number of measurements:

(3 ml + 3.4 ml + 7.2 ml + 3.7 ml + 4.3 ml) / 5 = 4.3 ml

Most people would stop here. However, there’s one very important piece of information missing: how accurate is that result? Surely an average value of 4.3 ml computed from 100 measurements is much more reliable than the same average computed from 5 measurements. Here’s where the standard error comes in and thanks to the internet, calculating it couldn’t be easier. You can type in the measurements here to get the standard error:

http://www.miniwebtool.com/standard-error-calculator/

It tells us that the standard error (of the mean, to be pedantically precise) of my five measurements is SEM = 0.75. This number is extremely useful because there’s a rule in statistics that states that with a 95 % probability, the true average lies within two standard errors of the computed average. For us this means that there’s a 95 % chance, which you could call beyond reasonable doubt, that the true average of my daily liquid consumption lies in this intervall:

4.3 ml ± 1.5 ml

or between 2.8 and 5.8 ml. So the computed average is not very accurate. Note that as long as the standard deviation remains more or less constant as further measurements come in, the standard error is inversely proportional to the square root of the number of measurements. In simpler terms: If you quadruple the number of measurements, the size of the error interval halves. With 20 instead of only 5 measurements, we should be able to archieve plus/minus 0.75 accuracy.

So when you have an average value to report, be sure to include the error intervall. Your result is much more informative this way and with the help of the online calculator as well as the above rule, computing it is quick and painless. It took me less than a minute.

A more detailed explanation of the average value, standard deviation and standard error (yes, the latter two are not the same thing) can be found in chapter 7 of my Kindle ebook Statistical Snacks (this was not an excerpt).