Σ x i. Sigma Notation

Transcription

1 Sigma Notation The mathematical notation that is used most often in the formulation of statistics is the summation notation The uppercase Greek letter Σ (sigma) is used as shorthand, as a way to indicate that a sum is to be taken: Σ x i The notation above is equivalent to writing x 1 + x 2 + x x n-1 + x n

2 Sigma Notation - Components Looking at sigma notation in more detail: refers to where the sum of terms ends indicates we are taking a sum Σ x i indicates what we are summing up refers to where the sum of terms begins

3 Sigma Notation - Indices For example, with the following indices: i=5 Σ x = x i 3 + x 4 + x 5 i=3 The above sum indicates that only observations 3 through 5 should be included in the sum, despite the fact that x 1 through x n might exist in the sample x 1 + x 2 + x x n-1 + x n

4 Sigma Notation - Rules Constants (denoted by an a in the example below) in summation notation can be treated as follows: n times Σ a = (a + a + a) = na Here we are simply using the summation notation to carry out a multiplication, e.g. i=3 Σ 4 = (4+4+4) = 4*3 = 12

5 Sigma Notation - Rules If we have a constant within a summation, that constant can be taken outside the summation sign: Σ ax i = a Σ x i Σ ax i = ax 1 + ax ax n a Σ x i = a(x 1 + x x n )

6 Sigma Notation - Rules The order in which addition operations are carried out is unimportant: Σ (x i + y i ) = Σ x i + Σ y i (x 1 + x 2 + x x n-1 + x n ) + (y 1 + y 2 + y y n-1 + y n )

7 Sigma Notation - Rules Exponents are handled differently depending on whether they are applied to the observation term or the whole sum: Σ x i m = x 1m + x 2m + + x m n ( Σ x ) m = (x i 1 + x x n ) m

8 Sigma Notation - Rules Products are handled much like exponents: Σ x i y i = (x 1 y 1 + x 2 y x n y n ) Σ x i * Σ x i y i = Σ x i * Σ y i Σ y i =(x 1 + x x n ) * (y 1 + y y n )

9 Sigma Notation - Simplification A summation will often be written leaving out the upper and/or lower limits of the summation, and in these cases we must assume that all of the terms available are to be summed: Σ x i = n Σ x i = i Σ x i = i Σ x i

10 Sigma Notation Compound Sums We frequently use tabular data (or data drawn from matrices), with which we can construct sums of both the rows and the columns (compounds sums), using subscript i to denote the row index and the subscript j to denote the column index: i rows j columns x 11 x 12 x 13 x 21 x 22 x 23 Σ j=m Σ x ij = (x 11 + x 12 + x 13 j=1 + x 21 + x 22 + x 23 )

11 Pi Notation Just as we have summation notation that we can use as a shorthand for sums, there is also product notation for multiplication The uppercase Greek letter Π (pi) is used as shorthand, as a way to indicate that a product is to be calculated: Π x i The notation above is equivalent to writing x 1 * x 2 * x 3 * * x n-1 * x n

12 Pi Notation While the product operator is less commonly applied in the statistical context, one place where it can be used is in the calculation of combinations, which make use of the factorial (!) operator:!n = n * n-1 * n-2 * * 3 * 2 * 1 = Π i!6 = 6 * 5 * 4 * 3 * 2 * 1 = 720 = i=6 Π i

13 Some Basic Terminology Population In statistical terms, a set of things (potentially any sort of things), but in particular referring to the complete set of all that are like them Variables The properties of a population that are to be measured (i.e. how do parts of the population differ?) Sample A subset (part of) the population Parent Population Refers to the more general population of origin for any given sub-population / sample / subset

14 Some Basic Terminology Representative A sample is representative if it is an accurate reflection of its parent population. Ensuring that samples are representative of the population from which they are drawn is a primary problem in statistics Constant Something that does not vary Parameter A constant measure which describes the characteristic of a population Statistic The corresponding measure for a sample

15 Some Common Corresponding Parameters and Descriptive Statistics (Population) Parameter (Sample) Statistic Mean µ x Standard Deviation σ S Variance σ 2 S 2

16 Simple Descriptive Statistics Descriptive statistics provide an organization and 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. Ratios 2. Proportions 3. Percentages 4. Rates of Change 5. (Location Quotients)

17 Simple Descriptive Summary Measures 1. Ratios - # of observations in A # of observations in B e.g. A - 6 overcast, B 24 mostly cloudy days 2. 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 or a color ={yellow, red, blue, green} a a count ={4, 6, 5, 2} proportion = i Σ a i

18 Simple Descriptive Summary Measures 2. Proportions cont. Sum of all proportions = 1 These 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 samples 3. Percentages Calculated by proportions x 100, e.g. 2/23 = 8.696%, use of these should be restricted to larger sample sizes, perhaps 20+ observations

19 Simple Descriptive Summary Measures 4. Rates of Change Expressing the change in a variable with respect to its original value, e.g. Z = x(t 2) x(t 1 ) x(t 1 ) = change in x original value of x e.g. if we had 20 marbles and then added 10, the rate of change = (30-20)/20 = 10/20 = Location Quotients An index of relative concentration in space, a comparison of a region s share of something to the total

20 Simple Descriptive Summary Measures 5. Location Quotients cont. For example, suppose we have a region of 1000 km 2 which we subdivide into three smaller areas of 200, 300, and 500 km 2 respectively (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 Quotient A 200/1000= /600= /0.2=1.25 B 300/1000= /600= /0.3=0.57 C 500/1000= /600= /0.5=1.17 Location Quotient = Prop. of Cases / Prop. of Area

21 Simple 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 tendency

22 Measures 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. Mode 2. Median 3. Mean Each of these measures is appropriate to different distributions / under different circumstances

23 Measures of Central Tendency - Mode 1. 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 quickly

24 Measures of Central Tendency - Median 2. 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)

25 Measures of Central Tendency - Median 2. Median cont. When we find the median of a distribution, we do so by dividing it into two equal parts. We can divide distributions into a greater number of parts as well: Quartiles each contains 25% of all values Quintiles each contains 20% of all values Deciles each contains 10% of all values Percentiles each contains 1% of all values Except for quintiles (because they give an odd number of parts) each of these also gives us the median

26 Measures of Central Tendency - Mean 3. Mean a.k.a. average, the most commonly used measure of central tendency x = Σ x i n Sample mean µ = i=n Σ x i N Population mean When we compute a mean using these basic formulae, we are assuming that each observation is equally significant

27 Measures of Central Tendency - Mean 3. Mean cont. We can also calculate a weighted mean using some weighting factor: x = Σ w i x i Σ w i e.g. What is the average income of all people in cities A, B, and C: City Avg. Income Population A $23, ,000 B $20,000 50,000 C $25, ,000 Weighted mean Here, population is the weighting factor w i and the average income is the variable of interest x i

28 Measures of Central Tendency - Mean 3. Mean cont. We can also calculate a grouped mean using the mid-points and frequencies of groups: x = Σ m i f i Σ f i Grouped mean e.g. Suppose we had grouped some data in a frequency table and wanted to calculate the grouped mean: Group Freq. Midpoint

29 Measures of Central Tendency - Mean 3. Mean cont. A standard geographic application of the mean is to locate the center (a.k.a. centroid) of a spatial distribution by assigning to each member of the spatial distribution a gridded coordinate and calculating the mean value in each coordinate direction Bivariate mean or mean center For a set of (x,y) coordinates, the mean center (x,y) is computed using: x = Σ x i n y = Σ y i n

30 Measures of Central Tendency - Mean 3. Mean cont. We can also calculate a weighted mean center in much the same way, but by using weights: For a set of (x,y) coordinates, the weighted mean center (x,y) is computed using: x = Σ w i x i Σ w i y = Σ w i y i Σ w i e.g. suppose we had the centroids and areas of 3 polygons Here we weight by area, but other weightings possible

31 Measures of Central Tendency - Mean 3. Mean cont. NOTE: Mean centers and weighted mean centers are very sensitive to outliers! For example: Suppose we calculated the geographic center of the United States when considering all states. Consider the influence that Hawaii and Alaska would have on the result if we chose to include them in the calculation. Clearly we would get a very different result with and without these two states

32 Selecting a Measure of Central Tendency Most often, the mean is selected by default The mean s key advantage is that is sensitive to any change in the value of any observation However, we really must consider the nature of the distribution to choose properly: 1. Multi-modal distribution The mode must be used, as the median or mean would be rather meaningless 2. Unimodal symmetric The mean is fine if the distribution approaches being symmetric

33 Selecting a Measure of Central Tendency 3. Unimodal skewed A skewed distribution creates significant differences between the measures of central tendency: Negatively skewed Mean < Median < Mode Positively skewed Mode < Median < Mean In both cases of skew, the median is appropriate, especially in cases with extreme outliers (e.g. dist. of salaries of UNC Geography graduates) 4. Ordinal Data Median is well applied 5. Nominal Data Mode is the only choice