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1/3/2017 Measures of Location Summary Statistics: Measures of Location and Spread Illustrate where the majority of locations are found: e.g., means, medians, modes Ecological Analyses Ecological Analyses Measures of Spread Illustrate Statistics versus Parameters how variable the data are: Statistics describe the sample Parameters describe the [unknown?] population e.g., standard deviation, variance, standard error Ecological Analyses Ecological Analyses Measures of Location: Mean Arithmetic Mean Unbaised Arithmetic estimate of if: Observations from random individuals Samples are independent of eachother Observations drawn from a large population that can be described by a normal random variable Mean All observations weighted equally in calculation X~N(,) Ecological Analyses Ecological Analyses 1 1/3/2017 Bias and Sample Size Other Means Geometric Mean Example from exponential population growth: when numbers are multiplied on an arithmetic scale then can be added on a logarithmic scale ... So it depends how you use the ‘mean’ Ecological Analyses Ecological Analyses Other Means Harmonic Median and Mode Mean Uses the reciprocal (1/Y) of the data Sensitive to extreme values that are small Often in conservation biology: effective population size Ecological Analyses Median and Mode Ecological Analyses Which measure of location? Arithmetic Median: the ‘middle’ observation (unless tied) Mode: the observations that occurs most frequently mean most common Supported by Central Limit Theorem Geometric mean most appropriate for multiplicative measures Median or Mode when distribution doesn’t match a standard probability distribution Ecological Analyses Ecological Analyses 2 1/3/2017 Which measure of location? Measures of Spread Researchers use different measure of location to support different points of view... Pay attention to what measure is supplied and always be suspicious of any measure of location that is not accompanied by a measure of spread! Variance Sum and Standard Deviation of Squares (SS): Ecological Analyses Ecological Analyses Measures of Spread Degrees of Freedom Variance The number of independent observations that we have for estimating statistical parameters For now ... n-1 Unbiased estimate of 2 Ecological Analyses Measures of Spread Ecological Analyses Standard Error of the Mean Variance Think Standard of the standard error (or the mean) as an estimate of the standard deviation of the POPULATION MEAN Deviation Ecological Analyses Ecological Analyses 3 1/3/2017 Standard Error of the Mean Standard Error of the Mean If inference is about the sample: provide SD (s) If the inference is about the means: provide the SE Ecological Analyses Skewness, Kurtosis, and Central Moments A central moment is the average of the deviations of all observations in a dataset from the mean of the observations, raised to a power r: Ecological Analyses Skewness, Kurtosis, and Central Moments r = 1 (1st moment) always 0 r = 2 (2nd moment) is the variance Ecological Analyses Ecological Analyses Skewness Skewness g1 = r = 3 (3rd moment) divided by s3 = skewness 0 normal distribution g1 > 0 right-skewed (longer tail of observations to the right of the mean Skewness describes how the sample differs in shape from a symmetrical distribution Ecological Analyses g1 < 0 left-skewed (longer tail of observations to the left of the mean Ecological Analyses 4 1/3/2017 Skewness Kurtosis Based on 4th central moment (r=4) Measures the extent to which the distribution is distributed in the tails versus the center of the distribution Ecological Analyses Ecological Analyses Kurtosis Kurtosis Clumped, or platykurtic distributions have g2 < 0 (less probability in the tails) Leptokurtic distributions have g2 > 0 (less probability in the center) Ecological Analyses Ecological Analyses Skewness and Kurtosis Quantiles Should be tested, but both measures are sensitive to outliers ... Ecological Analyses Box plots of quantiles can portray the distribution of data more accurately than plots of means and standard deviations Ecological Analyses 5 1/3/2017 Other Measures Distribution of Points For normally distributed random variables: Coefficient of Variation (CV) Variability ‘independent of the mean’ Coefficient of Dispersion For discrete variables ‘varianceto-mean’ ratio Measure of clumping, but dependent on ‘scale’ 67% of observations occur within 1 SD of the mean 96% of observations occur within 2 SD of the mean Ecological Analyses Confidence Intervals Ecological Analyses Confidence Intervals Ecological Analyses Confidence Intervals Ecological Analyses Confidence Intervals Ecological Analyses Ecological Analyses 6 1/3/2017 Confidence Intervals Confidence Intervals Interpretation: 95% of the time such an interval will contain the true value of NOT: “there is a 95% chance that the true occurs within the interval” it either does or does not ... Ecological Analyses Ecological Analyses 7