Simple Descriptive StatisticsSimple Descriptive Summary MeasuresSimple Descriptive Summary MeasuresSimple Descriptive Summary MeasuresSimple Descriptive Summary MeasuresSimple Descriptive StatisticsMeasures of Central TendencyMeasures of Central Tendency - ModeMeasures of Central Tendency - MedianMeasures of Central Tendency - MeanMeasures of Central Tendency - MeanMeasures of Central Tendency - MeanMeasures of Central Tendency - MeanMeasures of Central Tendency - MeanWhy Do We Need Measures of Dispersion at all?Measures of Dispersion - RangeMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresMeasures of Dispersion – Variance, Standard Deviation, Z-scoresFurther Moments of the DistributionFurther Moments of the DistributionFurther Moments of the DistributionFurther Moments of the DistributionFurther Moments of the DistributionFurther Moments of the Distribution - SkewnessFurther Moments of the Distribution - KurtosisFurther Moments of the Distribution - KurtosisDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Statistics• Descriptive statistics provide an organizationand summary of a dataset•A small number of summary measures replaces the entirety of a dataset• You’re likely already familiar with some simple descriptive summary measures:1. Ratios2. Proportions3. Percentages4. Rates of Change5. (Location Quotients)David Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Summary Measures1. Ratios - # of observations in A# of observations in Be.g. A - 6 overcast, B – 24 mostly cloudy days2. Proportions – Relates one part or category of data to the entire set of observations, e.g. a box of marbles that contains 4 yellow, 6 red, 5 blue, and 2 green gives a yellow proportion of 4/17 oracolor={yellow, red, blue, green}acount={4, 6, 5, 2} proportion = aiΣaiDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Summary Measures2. Proportions cont. – Sum of all proportions = 1These are useful for comparing two sets of data w/ different sizes and category counts e.g. a different box of marbles gives a yellow proportion of 2/23, and in order for this to be a reasonable comparison we need to know the totals for both samples3. Percentages – Calculated by proportions x 100, e.g. 2/23 = 8.696%, use of these should be restricted to larger sample sizes, perhaps 20+ observationsDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Summary Measures4. Rates of Change – Expressing the change in a variable with respect to its original value, e.g.Z =x(t2) –x(t1)x(t1)change in xoriginal value of x=e.g. if we had 20 marbles and then added 10, the rate of change = (30-20)/20 = 10/20 = 0.55. Location Quotients – An index of relative concentration in space, a comparison of a region’s share of something to the totalDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Summary Measures5. Location Quotients cont. – For example, suppose we have a region of 1000 km2which we subdivide into three smaller areas of 200, 300, and 500 km2respectively (labeled A, B, and C)• The region has an influenza outbreak with 150 cases in the first region, 100 in the second, and 350 in the third (a total of 600 flu cases):Proportion of Area Proportion of Cases Location QuotientA 200/1000=0.2 150/600=0.25 0.25/0.2=1.25B 300/1000=0.3 100/600=0.17 0.17/0.3=0.57C 500/1000=0.5 350/600=0.58 0.58/0.5=1.17Location Quotient = Prop. of Cases / Prop. of AreaDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Simple Descriptive Statistics• These are ways to summarize a number set quickly and accurately• The most common way of describing a variable distribution is in terms of two of its properties:• Central tendency – describes the central value of the distribution, around which the observations cluster• Dispersion – describes how the observations are distributed• First we’ll look at measures of central tendencyDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Measures of Central Tendency• Think of this from the following point of view: We have some distribution in which we want to locate the center, and we need to choose an appropriate measure of central tendency. We can choose from:1. Mode2. Median3. Mean• Each of these measures is appropriate to different distributions / under different circumstancesDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Measures of Central Tendency - Mode1. Mode – This is the most frequently occurring value in the distribution• In the event that multiple values tie for the highest frequency, we have a problem …• A potential solution in this situation involves constructing frequency classes and identify the most frequently occurring class• This is the only measure of central tendency that can be used with nominal data• The mode allows the distribution’s peak to be located quicklyDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Measures of Central Tendency - Median2. Median – This is the value of a variable such that half of the observations are above and half are below this value i.e. this value divides the distribution into two groups of equal size• Note: When the distribution has an even number of observations, finding the median requires averaging two numbers•The key advantage of the median is that its value is unaffected by extreme values at the end of a distribution (which potentially are outliers)David Tenenbaum – GEOG 070 – UNC-CH Spring 2005Measures of Central Tendency - Mean3. Mean – a.k.a. average, the most commonly used measure of central tendencyΣxii=1i=nnx =Sample meanΣxii=1i=NNµ=Population mean• When we compute a mean using these basic formulae, we are assuming that each observation is equally significantDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Measures of Central Tendency - Mean3. Mean cont. – We can also calculate a weighted mean using some weighting factor:Σwi xii=1i=nx =Weighted meanΣwii=1i=ne.g. What is the average income of all people in cities A, B, and C:City Avg. Income PopulationA $23,000 100,000B $20,000 50,000C $25,000 150,000Here, population is the weighting factor