Just what is normal, anyway?

Yesterday's blog post talked about travelling the country, visiting places that had a normal monthly temperature of 70 degrees. According to the National Centers for Environmental Information:

What Are Climate Normals?

Scientists traditionally define a Climate Normal as an average over a recent 30-year period. Our most recent installment covers the period from 1981 to 2010. Why 30 years? Close to a century ago, the International Meteorological Organization—now known as the World Metrological Organization (link is external)—instructed member nations to calculate Climate Normals using 30-year periods, beginning with 1901–1930. Also, a general rule in statistics says that you need at least 30 numbers to get a reliable estimate of their mean or average. So, our scientists have traditionally defined Normals as averages over 30 years simply because that is the accepted convention—not because a 30-year average is the only logical or “right” way to define a Climate Normal.

A bell shaped curve is called a normal curve and according to the Central Limit Theorem in statistics:

The Central Limit Theorem (CLT for short) basically says that for non-normal data, the distribution of the sample means has an approximate normal distribution, no matter what the distribution of the original data looks like, as long as the sample size is large enough (usually at least 30) and all samples have the same size. And it doesn’t just apply to the sample mean; the CLT is also true for other sample statistics, such as the sample proportion. Because statisticians know so much about the normal distribution, these analyses are much easier. 

So, your data can approximate a normal curve if you have at least 30 data items.

According to their data, the "normal" high temperature for October 17th in Modesto, CA is 78.7 and the predicted high was 82 degrees. As we talked about on Monday, the average, or mean, gives only half the information. We also need the standard deviation, which is not included with the normal temps. But since our high today is only a little more than 3 degrees from the mean, we can probably safely assume that it's a reasonably "normal" value.

The normal temperature for today (it's the 17th as I'm writing this) was found by taking the average of the high temerature on this day every year between 1981 and 2010. The normal high for the month is the average of the high temperatures for each day of the month. Here are the monthly "normal" highs for Modesto:

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
55.4	63.8	68.7	74.7	82.9	89.9	95.1	93.6	89.0	79.0	65.5	55.6

The average annual temperature is 76.1 degrees and the standard deviation is 14.3 degrees, which says that about two-thirds of the months should have an average high temperature between 61.8 and 90.4 degrees, which they do. We can say something like "The monthly high temperatures range from a low of 55 degrees in January, to a high of 95 degrees in July. Eight months of the year have an average high temperature between 62 and 90 degrees".

Meanwhile, in International Falls, MN, the average annual temperature is 48.7 degrees and the standard deviation is 23.1 degrees, which says that about two-thirds of the months should have an average high temperature between 25.6 and 71.8 degrees, which they don't.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
15.4	22.0	34.7	51.5	64.8	73.2	77.8	75.9	65.4	51.1	33.7	19.0

We can say something like "The monthly high temperatures range from a low of 15 degrees in January, to a high of 78 degrees in July. Six months of the year have an average high temperature between 26 and 72 degrees". In this case, the median would make a better measure. The median is 51.3, Q1 is 30.8 and Q3 is 67.4. Fifty percent of the values will fall in between Q1 and Q3. Or 6 values should fall between 31 and 67 degrees, which they do. Since there are only 12 data items, we don't really have a normal distribution in either case. It's just that Modesto was a little better behaved. Here's a graph of the monthly high temperatures, with Modesto in red and International Falls in blue.



So, now you know what normal means. At least as far as temperature goes.

Interesting days

Today - Chocolate Cupcake Day, Hagfish Day, Global Dignity Day and No Beard Day

Tomorrow - Get Smart About Credit Day, Conflict Resolution Day and Evaluate Your Life Day

Next Wednesday - International Artists Day, Sourest Day, Punk for a Day Day, Greasy Food Day and World Pasta Day

November 18 - Occult Day, Housing Day, Mickey Mouse Day and GIS Day