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Statistics

- Statistics deals with the collection, processing,

analysis and interpretation of numerical data. - It is critical for the credibility of any data

collection and interpretation to utilize proper

statistics. - Biology
- Meteorology
- Oceanography

Statistics - definitions

- Statistics the mathematics of the collection,

organization, and interpretation of numerical

data, especially the analysis of population

characteristics by inference from sampling - Population the set containing all possible

elements of interest in a study - Sample a subset of a population
- Inferential Statistics using information

obtained from a sample to draw conclusions about

a population (one infers meaning from the sample

and projects it on the population as a whole. - Frequency Number of times an event occurs

Statistics - definitions

- Frequency Distributions A table that divides a

set of data into a suitable number of classes

(categories or intervals), showing the number of

times an observation fits into that class

(category or interval). It is important to

consider the interval size when creating a proper

frequency distribution to ensure one captures the

details of the data. - ---As a general rule of thumb, let the number of

intervals equal 5log(N) where N is the size of

the sample or population1. - 1 H. Panofsky and G. Brier, Some Applications

of Statistics to Meteorology, Earth and Minearl

Sciences Continuing Education, University Park,

PA. 1968. Pg 4.

Frequency Distribution

- Frequency distribution of the number of rain-days

in the month of January Camden square London over

the years 1858-1947

Histogram

- Histogram - A graphical display of a frequency

distribution. A histogram is created by

allocating the data in various bins.

Pie Chart

- Pie chart A circular chart divided into sectors

indicating the relative frequency or percent of a

specific sample. - Pie chart showing the distribution of water and

salts in seawater as well as the breakdown of the

different salinity ions of various salts in

seawater.

Measures of center

- Measures of center provide an indication of the

central value of the dataset. - Mean the arithmetic average value of a sample

(sum of the observations divided by the number of

observations). For a sample of size N, it is

mathematically defined as

Measures of center

- Median the middle value (the central

observation) of an ordered sample. It is

mathematically defined as follows - Mode The most frequent value in the dataset.

For a given data set, if a two value occur as the

most frequent, then the sample is considered

Bi-modal with both values considered viable

modes. This reasoning extends to higher order

modes as well.

Measures of spread

- Measure of spread indicate how much variation

exists in the dataset. - Quartiles if a distribution is divided into

four equal classes, the quartiles (Q1, Q2, Q3,

and Q4), represent the observations in which 25,

50, 75 and 100 of the observations lie below

the values of the quartiles.

Measures of spread

- Percentiles - if a distribution is divided into

100 equal classes, the percentiles (P.01, P.02,

and P1.0), represent the observations in which x

of the observations lie below the values of the

percentiles. This can be considered a refinement

of the quartile measure of spread introduced

above. -

Measures of spread

- Variance (s2) the average of the squared

deviations from the mean. For a sample size N,

the variance is mathematically defined as - Standard Deviation (s) the average deviation

from the mean for a sample. It is mathematically

defined by taking the square root of the variance.

Normal distributions

- One of the most common distributions is the

normal distribution. - This distribution is apparent in distributions

where most values lie near a central value and

deviations from the commonly accepted average

fall off in a proper functional form. Some

examples include - - population frequency
- - grade distribution
- - Climatology parameters
- The normal or Gaussian distribution is defined by

the mathematical function

Normal distributions

- A graph of a normal distribution is shown below

for - and . Notice how the

graph is more spread out and shorter.

Normal distributions - properties

- I. The mean median mode in a normal

distribution. - The converse of the above however is not

necessarily true though as seen by the following

graph where the mean median mode but is not

Gaussian.

Normal distributions - properties

- II. Approximately 68 of the observations lie

between (within one standard deviation of the

mean). - Approximately 95 of the observations lie between

(within two standard deviations of the mean). - Approximately 99 of the observations lie between

(between three standard deviations of the mean).

Skewness

- Skewness is a measure of the asymmetry of a

sample distribution about the mean. - A graph indicating a perfectly symmetric normal

distribution about the mean and a

distribution where the mean is clearly skewed to

the left.

Linear Correlations

- Often one wishes to determine how linear one set

of data correlates to another. In other words,

given two sets - One wishes to quantify how well the following

relationship exists between the two sets - Where a and b are suitably chosen constants that

can be found by the use of simple linear

regression.

Pearson Product-moment correlation coefficient

(MCV)

- The Pearson Product-moment correlation

coefficient or MCV is a common way to determine

how well the data is linearly correlated. - THE MCV is denoted by the variable rxy and has an

absolute value of 1 if the data is perfectly

linearly correlated. If the data has absolutely

no linear correlation than rxy0 and the data

is said to be randomly correlated - A large correlation can either refer to a slope

that can be either positive or negative. The

corresponding value of rxy will either be near

1 or -1 depending on if the slope is greater

than 0 or less than 0 respectively.

Pearson Product-Moment Correlation Coefficient

(MCV)

- The mathematical definition of the MCV for a

sample N is - Graph with random variations but an MCV of

rxy0.98

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