Statistical Summary ATM 305 – 12 November 2015. Review of Primary Statistics Mean Median Mode x i...

Statistical Summary

ATM 305 – 12 November 2015

Review of Primary Statistics

• Mean

• Median

• Mode

xi - scalar quantityN - number of observations

Value at the 50% cumulative frequency distribution

Ex - 8, 25, 35, 60, 85, 90, 115, 200, 82550% 50%

Maximum (or maxima) value of a distributionMultiple modes possible (maxima)

y

x

mode

Wind• Mean wind speed

– Mean speed determined without consideration of direction of wind

• Mean resultant wind- vector wind

To calculate it is necessary to compute the individual ui and vi wind components, average them separately, and recombine and to get the mean vector wind.

Note the differences between mathematical wind direction and meteorological wind direction

SW wind

Let = meteorological wind direction= 225o

= 45o

= 270o –

= 270o –

Wind• Mean resultant wind (cont.)

Using mathematical wind direction

Mean


Using meteorological wind direction

Resultant Wind Speed Resultant Wind Direction

Mean

Note Changes


To get meteorological wind direction right, necessary to check individual coordinates

u

v> 0

> 0

> 0

< 0

< 0

> 0

< 0

< 0

a) > 0 and > 0 (SW winds)

b) < 0 and < 0 (NE winds)c) > 0 and < 0 (NW winds)

d) < 0 and > 0 (SE winds)

e) = 0 and > 0 (S winds)

f) = 0 and < 0 (N winds)g) > 0 and = 0 (W winds)

h) < 0 and = 0 (E winds)

a)

b)

c)

d)

= 180o

= 0o

= 270o

= 90o


Resultant wind speed is smaller than average speed of individual winds

The mean wind speed (NOT resultant wind) determines mean evaporation, cooling power, and boundary layer turbulence

Example: 10 wind measurements

5 northerly wind measurements, each at 10 kt5 southerly wind measurements, each at 10 kt

What is the mean wind speed, and resultant wind speed?

= (0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0) / 10 = 0

= (-10) + (-10) + (-10) + (-10) + (-10) + 10 + 10 + 10 + 10 + 10) / 10 = 0

mean wind speed = (10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10) / 10 = 10 kt

resultant wind speed

= 0 kt

Wind• Wind Persistence (“steadiness”)

P = 1 Winds always blows from the same direction (resultant wind and mean wind speed are equal) P = 0 Wind is unsteady and blowing from all directions

Standard Deviation

Sometimes denoted by sigma,

Note:

Variance:

Note: some texts when N is large replace N with (N – 1)

Both the standard deviation and variance measure the spread of a large sample of data

Simply the square of standard deviation

Mean AnomalyIndividual Value

• Standardizing anomalies allows us to compare anomalous values in different parts of the globe

• Can be thought of as a measure of distance in standard deviation units of a value from its mean

All that is needed is a mean and standard deviation and you can find the standardized anomaly of a variable

Anomaly

Standard Deviation

Individual Value

Mean of that value

Example: standardized anomalies of 500-hPa geopotential heights

See Link:

• Can now directly compare low heights over midwestern US to high heights over east Pacific

• Standard deviation of heights relatively low in tropics, but relatively high in mid-latitudes

http://www.tropicaltidbits.com/analysis/models/?model=gfs&region=atl&pkg=z500a_sd&

Anomalies

Standardized Anomalies




• Standardizing anomalies allows us to compare anomalous values in different parts of the globe

• Can be thought of as a measure of distance in standard deviation units of a value from its mean

All that is needed is a mean and standard deviation and you can find the standardized anomaly of a variable

Standardizing a dataset results in a mean = 0 and a standard deviation = 1

Anomaly

Standard Deviation

Individual Value

Mean of that value

Example: standardized anomalies of 500-hPa geopotential heights

• Can now directly compare low heights over midwestern US to high heights over east Pacific

• Standard deviation of heights relatively low in tropics, but relatively high in mid-latitudes

See Link: http://www.tropicaltidbits.com/analysis/models/?model=gfs&region=atl&pkg=z500a_sd&

Standardized anomalies




• Southern Oscillation Index (SOI)

Additional applications of standardized anomalies

Teleconnection that measures the standardized monthly sea level pressure difference between Tahiti and Darwin, Australia

Often used as a proxy for ENSO- Positive SOI corresponds to La Nina like conditions- Negative SOI corresponds to El Nino like conditions

Link: https://www.ncdc.noaa.gov/teleconnections/enso/indicators/soi/

El Nino

La Nina

https://www.ncdc.noaa.gov/teleconnections/enso/indicators/soi/

Normal Distribution• Also known as a Gaussian distribution or “bell curve”

Amplitude

Standard Deviation

-2 -1 0 1 2

Only 2.5% of distribution

most meteorological variables have normal distributions

- Temperature, Geopotential Height(these fields are good to “standardize”)

Some meteorological variables DO NOT have normal distributions

- Precipitation, precipitable water, windspeed

68%

95%

Cumulative Frequency Distributions• Depicts the cumulative total of a variable over time

CumulativePrecipitation

Normal precipitation

Time

Jan 1 Dec 31

Below Normal

Above Normal

Daily Precipitation

http://www.cpc.ncep.noaa.gov/products/global_monitoring/precipitation/global_precip_accum.shtml

URL to CFD plots from Climate Prediction Center:



Pearson (Ordinary) CorrelationCorrelation measures the strength of a linear relationship between 2 variables

Alternative Computational Form (useful if you don’t have standardized anomalies)

Note: -1 ≤ rxy ≤ 1-1,+1 Indicates perfect linear association between the two variables0 Indicates no linear relationship between the two variables

Pearson (Ordinary) CorrelationPlot Examples

Y va

riabl

e

X variable X variable X variable

Y va

riabl

e

Y va

riabl

er ≈ 0r = 1 r = -1

Note: even if there is no linear relationship, a strong non-linear relationship can exist!

IMPORTANT: Correlation can suggest causation, pending a physical explanation of the relationship, but it can NEVER, by itself, imply causation!

Simple Linear RegressionLet some variable x be the independent or predictor variable (i.e., SSTs)Let some variable y be the dependent or predictand variable (i.e., # of TCs)- A linear regression procedure chooses the line producing the least

error for predictions of y given x, using a least squares approach

- Linear regression can be used to predict linear changes in variables, and is often an important component of statistical models.

- In calculus we write a line as:- Stasticians write line as:

y-intercept slope

Use this form if you already know correlation coefficient

see youtube video explanation: https://www.youtube.com/watch?v=jEEJNz0RK4QCool visualization of least squares approach

http://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

https://www.youtube.com/watch?v=jEEJNz0RK4Q

http://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

Simple Linear RegressionSimple example

xi yi

-2 0-1 00 11 12 3 -2 -1 0 1 2

0

1

2

3

4

X-AxisY-

Axi

sxi*yi xi

2

0 40 10 01 16 4

- Sum these values

- Solve for b

= 0.7b

Simple Linear RegressionSimple example

xi yi

-2 0-1 00 11 12 3 -2 -1 0 1 2

0

1

2

3

4

X-AxisY-

Axi

sxi*yi xi

2

0 40 10 01 16 4

- Sum these values

- Solve for a

= 1

Correlation and Regression Maps

Available on NOAA ESRL Map Room: http://www.esrl.noaa.gov/psd/data/correlation/

Correlation

Example: Correlation of NAO with 500-hPa geopotential heights

Strong negative correlation

NAO

500-

hPa

Hgt r = -0.8

More positiveMore negative

Low

HighAt each grid point in Northern Hemisphere

Coast of Greenland

http://www.esrl.noaa.gov/psd/data/correlation/



Correlation


Moderate positive correlation

NAO50

0-hP

a H

gt

r = +0.6


Low

High

At each grid point in Northern Hemisphere Eastern US




Correlation


No Correlation

NAO50

0-hP

a H

gt

r = ≈ 0


Low

High

At each grid point in Northern Hemisphere

Central Pacific




Regression

Example: Regression of NAO with 500-hPa geopotential heights

At each grid point in Northern Hemisphere

Note that Regression and Correlations maps show the same features, but amplitude of regression much greater than correlation which must be between -1 to 1

NAO

500-

hPa

Hgt


Low

HighCoast of Greenland

regression line: Amplitude (≈ -60 m)


Simplified Statistical Summary PlotsBox and Whisker Display

Max

Upper quartile

Min

Lower quartile

MedianBoxWhiskers

Simplified Statistical Summary PlotsHistogram

Bins (amounts)

# of

eve

nts

0-10 10-20 20-30 30-40

Statistical Summary ATM 305 – 12 November 2015. Review of Primary Statistics Mean Median Mode x i...

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