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Reading And Writing About Statistics

Why do we need statistics? Statistics provides a standardized procedure for drawing conclusions. It is involved with the collection, analysis, interpretation, and presentation of data.

Distribution: If the individual subjects in a sample or population have a characteristic that is not the same among members of the sample, the description of that characteristic is the distribution.

Kinds of Data and Scales of Measurement

Data Type	Explanation and Examples
Nominal Scale	Grouping is arbitrary. Observations can be separated according to category: Dodge/Chevy/Ford, Male/Female. Classifications are equally weighted and sorted of data is only into named groups. All nominal scales are made from discrete data.
Ordinal Scale	Items can be ranked from smallest to largest. You can guess the relative weights of animals and rank them from smallest to largest: Shrew-Cat-Dog-Horse-Elephant All ordinal scales are composed of discrete data
Interval Scale	Based on continuous data. The numerical value has meaning. Example: A temperature of 100° C is exactly half that of 200° C. In the ordinal scale, the difference in weight among each of our animals is unknown. Interval scales my be either interval (discrete) or continuous (real).

Descriptive Statistics

Arithmetic Mean (): The average of the observations.

Median: The median is a statistic of location equal to that variable in a ranked array that has an equal number of items above and below it. The median divides the distribution in half. For the ranked distribution 14, 15, 16, 19, 23 M=16. When the sample has an even number of items, the median is the midpoint between the two sample variates (n/2)^th and [(n/2)+1]^th such as 14, 15, 16, 19 M=15.5. Use the median if the data are not normally distributed. An example might be the salaries in a company where a few high-paid executives will skew the mean salary.

Statistics of Dispersion

Figure 1

Radically different distributions can have the same mean (Figure 1). Therefore, measures of the dispersion is needed.
Range: The difference between the highest and lowest values. For the following distribution... 14.9, 10.8, 12.3, 23.3, the range is 23.8-10.8=12.5
Variance (s, σ², S): Takes the distribution into account and weights each variate by their distance from the mean. The variance is the mean of the squares of the differences between each sample and their mean, expressed mathematically as s²=(1/n)[S(x_i-x)²].

Figure 2

Standard Deviation (s, σ, SD): The standard deviation is the square root of the variance. Figure 1 depicts two distributions with equal means but low and high standard deviations. More on the standard deviation can be found here. Specifically, 68% of the observations lie within one standard deviation of the mean and 95% of the observations lie within two standard deviations of the mean (Figure 2).
Standard Error (SE, SEM): SD/sqrt(N).
95% Confidence Limits: If an experiment were run 100 times and you calculate the mean you would expect it to fall within the 95% confidence limits 95% of the time. For samples over 15, the 95% confidence limits are approximately twice the SEM

Example: Rat tail lengths (cm)

	Sample A	Sample B
	7.2	3.6
	7.0	12.5
	6.8	3.3
	7.0	8.6
N	4	4
	7.0	7.0
Median	7.0	7.0
s²	0.27	19.53
s	0.16	4.40

Skewness: Measures of how the data "leans". One tail is longer than the other. (Figure 3)
Kurtosis: Measure of the "peakiness" compared to a normal distribution. (Figure 3)

Figure 3

EXAMPLE DATA

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