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The Logarithmic Scale

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The Logarithmic Scale IIEEEEROO nnlliinneCC llaassssrroooom| LL ooggaarriitthhmmiicSS ccaallee hhttttpp::////wwwwww..iieeeerr..oorrgg//ccllssssrroooomm//lloogg..hhttmmll IEER | | Online Classroom The Logarithmic Scale This doesn't refer to tapping out a beat on tree stumps, but it can be just as fun! If we want to plot something that changes with time and the time period is relatively short, we often use a linear scale. Thus if we were considering 1,000 years, the linear scale might look like this: Each tick mark represents 100 years and each subdivision of the scale would be the same length. 0 100 200 300 400 500 600 700 800 900 1000 |______|______|______|______|______|______|______|______|______|______| TIME (YEARS) When we consider the many thousands of years it will be necessary to store nuclear waste in a geologic repository, it would not be possible to represent the decay on a linear time scale. So we resort to a convenient device called the logarithmic scale for plotting large numbers. In the logarithmic scale the only line segments that are equal are those that represent multiples by a constant factor, such as 2 or 3 or 10. So 1,000 years on a simple logarithmic scale that showed only the broad divisions would look like this:: 1 10 100 1000 |___________________|___________________|__________________| 101000 101011 101022 101033 TIME (YEARS) As you can see, such a scale can plot a great many more years than is possible on a linear scale, but its use would be limited by its lack of detail. However, if we were to divide each broad segment into nine segments and let the ticks represent the years from 1 to 10, 10 to 100, and 100 to 1000, the scale would look like this and would be much more useful: Note that each broad segment is subdivided in the same way. Each tick within a broad segment represents a multiple of 10 over the corresponding tick in the previous segment. For example, in the segment 1000 to 1011 the first mark equals 2. In the segment 10 11 to 1022 the first mark equals 20, and in the segment 1022 to 1033, the first mark equals 200. Notice that the line segment between 10 and 20 is equal to that between 20 and 40, which is equal to that between 40 and 80. Similarly the segment between 10 and 30 is equal to that between 30 and 90. That's all there is to it! Try this worksheet on logarithms if you would like to practice your new skills! On-Line Classroom Main Page Institute for Energy and Environmental Research Comments to: ieer [at] ieer.org Updated March 2, 2001, and May 5, 2006 Linear vs logarithmic scales. http://www.cs.sfu.ca/~tamaras/digitalAudio/Linear_vs_logarithmic.html Decibels CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Sound Power and Intensity Linear vs logarithmic scales. Human hearing is better measured on a logarithmic scale than a linear scale. On a linear scale, a change between two values is perceived on the basis of the difference between the values. Thus, for example, a change from 1 to 2 would be perceived as the same amount of increase as from 4 to 5. On a logarithmic scale, a change between two values is perceived on the basis of the ratio of the two values. That is, a change from 1 to 2 (ratio of 1:2) would be perceived as the same amount of increase as a change from 4 to 8 (also a ratio of 1:2). Figure 1: Moving one unit to the right increment by 1 on the linear scale and multiplies by a factor of 10 on the logarithmic scale. Decibels CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Sound Power and Intensity ``CMPT 889: Lecture 3, Fundamentals of Digital Audio, Discrete-Time Signals '' by Tamara Smyth , Computing Science, Simon Fraser University. Download PDF version (digitalAudio.pdf) Download compressed PostScript version (digitalAudio.ps.gz) Download PDF `4 up' version (digitalAudio_4up.pdf) Download compressed PostScrip `4 up' version (digitalAudio_4up.ps.gz) Copyright © 2005-10-06 by Tamara Smyth. Please email errata, comments, and suggestions to Tamara Smyth<[email protected]> School of Computing Science, Simon Fraser University The difference between the Linear and the Logarithmic Scales | The Analyt... http://www.morevisibility.com/analyticsblog/the-difference-between-the-l... The difference between the Linear and the Logarithmic Scales Subscribe Login or Regi http://www.morevisibility.com/analyticsblog/the-difference-between-the-linear-and-the-logarithmic- scales.html Recent Articles Share | Google Releases Googlebo March 4th, 2009 by Joe Teixeira Mobile: Part I By now, mostly everyone has tried using a Motion Chart by clicking on the “Visualize” hat Questions Can Googl button, toward the top of several Google Analytics reports at least one time. Analytics Answer? Part II Are Your Website Visitors S No! I didn’t even know it existed! What? What do you mean “ “? What do you mean “ “? Engaged? Well, if you haven’t taken a look, I strongly suggest doing so. For example, go to your Traffic hat Questions Can Googl Sources >> Keywords report and click on “Visualize” towards the top – you should see Analytics Answer? something like this: Google Analytics Individual Qualification Update Google Announcement: Site Speed Google Analytics Metrics an Dimensions Tutorial: Regular Expressio Google Analytics: The Pipe Symbol Google Announcement: Sec Search Topsy: Google Plus Search Article Categories Web Analytics Google Analytics There you go – you have just been visualized! . You can increase the number of circles Google Website Optimizer (or “bubbles”) that appear by going back to the regular report and increasing the number of Web Analytics Metrics rows, toward the bottom-right of the report table. A/B Testing One update to Motion Charts that happened a few months ago which flew a bit under the Multivariate Testing radar was the option for you to define your viewing scale – Linear, or Logarithmic. I can get Site Usability into a complex mathematical / statistical explanation of the differences between the two – Surveys / Polls which I’m sure will satisfy some of you – but for the masses, the easiest possible explanation I can use to differentiate the two is: MSN Gatineau Yahoo! Analytics Linear Scale - Based on Addition AW Stats Logarithmic Scale - Based on Multiplication IndexTools The first image above is an example of a Motion Chart using the Linear Scale. N otice on the Webside Story (HBX) X axis (going from left to right, t he “Pages / Visit” metric), the values from left to right Feedburner increase by 5 on each point – 5, 10, 15, 20, etc The values in the Y axis (Visits) increase by 100 on each point – 100, 200, 300, and so on. Google AdWords Key Performance Indicators Now let’s change “Lin” to “ Log” and watch what happens to our Motion Chart: Omniture SiteCatalyst Tealeaf WebTrends Coremetrics MSN Analytics Omniture Test & Target Urchin Software from Googl Competitive Intelligence Google AdSense ClickTracks NetInsight Mobile Google Plus Articles by Month The motion chart data is exactly the same, but much different looking now, is it not? Notice January 2012 how the bubbles are much more spread out and more “all over the place” using the December 2011 Logarithmic Scale. The Motion Charts in Google Analytics use what is known as a base 10 November 2011 sequence , as each point in the scale is multiplied by 10. A little tough to notice here in the X axis (Pages / Visit), but noticeable on the Y axis (Visits) – 1, 10, 100, 1,000, and it October 2011 continues off the chart infinitely. September 2011 The difference between the Linear and the Logarithmic Scales | The Analyt... http://www.morevisibility.com/analyticsblog/the-difference-between-the-l... When to use the Linear Scale? When to use the Logarithmic Scale? August 2011 July 2011 A good rule of thumb is to use the scale that best shows off your data. If you are only using the motion chart for the top 10 rows of your report, chances are that the l inear scale will June 2011 work just fine. If you’re going to be using 25 or more rows, you’ll most likely find it much May 2011 easier on the eyes if you use the logarithmic scale. So, less data = linear; more data = April 2011 logarithmic. But, please, play with the Motion Charts and decide for yourself what’s easier March 2011 for your chart. February 2011 “The Difference Between” Series: January 2011 The difference between Bounce Rate and Exit Percentage December 2010 Why are my visits different from my clicks? November 2010 Why are Conversions in Google AdWords different than Goals in GA? October 2010 The difference between Landing Pages and Top Content Pages September 2010 The difference between Google Analytics and Urchin Software from Google August 2010 Posted in Google Analytics, Web Analytics, Web Analytics Metrics | No Comments » | July 2010 May 2010 April 2010 Comments are closed at this time. March 2010 February 2010 January 2010 November 2009 October 2009 September 2009 August 2009 July 2009 June 2009 May 2009 April 2009 March 2009 February 2009 January 2009 December 2008 November 2008 October 2008 September 2008 August 2008 July 2008 June 2008 May 2008 April 2008 March 2008 February 2008 Related Sites Web Analytics MoreVisibility SEM Blog MoreVisibility SEO Blog MoreVisibility Social Media What is the difference between a logarithmic price scale and a linear one? http://www.investopedia.com/ask/answers/05/logvslinear.asp Home Dictionary Articles Tutorials Exam Prep Forex Markets Simulator FinancialE dge Free Too enterkeywords entersymbol Which penny stocks will rise? We'll tell you, free! 1.
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