## Thursday, May 26, 2016

### Unizor - Statistical Distribution - Task B - Problem

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Problem

In order to determine if the claims about climate change are true or false, the monthly average temperature data (in oF) at New York's Central Park have been gathered from the official Web site http://www.weather.gov/media/okx/Climate/CentralPark/monthlyannualtemp.pdf for the period from 1870 to 2015.

Based on these data, determine for each decade (10-year period) the mean, standard deviation and 95% certainty interval of average temperatures for each month of the year as well as for annualized average temperature.

Make a judgment about validity of the claims of New York climate change by comparing mean temperatures in different months between two decades, 1900's and 2000's separated by a century. Consider a comparison of a difference between averages with zero.

Solution

The raw data with some additional calculations in a spreadsheet format can be downloaded from http://www.unizor.com/Doc/MonthlyTempNYdata.xls.

This spreadsheet contains original data about average monthly temperatures for each month and annualized average temperatures during 1870-2015 period.

In addition, these data are supplemented with the following calculations for each month's and annualized temperatures for an entire period:

=MIN(...)

=MAX(...)

=SLOPE(...)

=AVERAGE(...)

=STDEV(...)

=VAR(...)

It is further supplemented with calculations used in one of the methods that determines validity of the climate change hypothesis based on sample variance of an average temperature per decade, assuming these variances do not change in time (see comment about it below, other approach is used as well).

These additional calculations include:

(a) variance of a decade average

=VAR(...)/10

(b) variance of a difference between two decade averages

= 2*(a)

(c) standard deviation of a difference between two decade averages

= SQRT(b)

(d) 95% margin of a difference between two decade averages

= 2*(c)

Finally, in the columns P and Q of this spreadsheet we have represented a line of linear regression (=TREND) of January (the coldest month) and July (the warmest month) temperatures.

As we see, there is a statistically visible trend in January and July temperatures of approximately 2.6oF (1.4oC) per 145 years or about 1oC per century. Other months have different trends, some less, some more (the highest increase of temperature is in March - more than twice that of January)

The degree of certainty of this conclusion is another issue, that we consider below.

The spreadsheet with original data and calculations per decade can be downloaded from http://www.unizor.com/Doc/MonthlyTempNY.xls.

Examine it. Here are some conclusions.

CONCLUSION

We can state with 95% certainty that decade average monthly temperature during 2000's is higher than the corresponding decade average monthly temperature during 1900's in February, April and August. November produced two different results in two methodologies we used. Annualized decade average temperature is also higher during 2000's than in 1900's.

The trend of about 1C (or slightly higher) per century is valid for at least four months and for an annualized temperature.

Obviously, we cannot state with any mathematical precision which factors contributed more or less to this increase. Politicians are making their careers fighting over these issues, which is completely outside this presentation.