A Short Course in Basic Statistics

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1 A Short Course in Basic Statistics Ian Schindler November 5, 2017 Creative commons license share and share alike BY: C 1 Descriptive Statistics 1.1 Presenting statistical data Definition 1 A statistical population is a set Ω of objects of study. We may assume that card(ω) <. Therefore there is a bijection i : Ω 1, n for some n N. This leads to using Ω and 1, n interchangeably. A random variable is a function X : Ω ω. The goal of descriptive statistics is to summarise the range of random variables Raw data Let X : Ω = 1, n ω. If ω R, the value of X at i 1, n, X(i) will be denoted x i. Continuous: ω = I R. Discrete: ω R and card(ω) is small. Categorical : ω is nominal A table of a simple statistical series For a discrete variable with K categories, {x i } n, we define: The class x k where x k < x k+1 for k 1, K 1. The number of elements in the class k: n k with n = K k=1 n k. The frequency: f k = n k /n with K k=1 f k = 1. 1

2 A discrete variable can be represented in a table: (k, x k, n k ) K k=1. (1) More completely: (k, x k, n k, f k ) K k=1 (2) The cumulative number of elements is s k = k l=1 n k with s K = n. The cumulative frequency Φ k = k l=1 f k. Below is a table representing a simple statistical series where Ω is 40 children and X : Ω R is their age in years. k x k n k s k f k Φ k ,1 0, ,2 0, ,35 0, ,30 0, ,05 1 The table can be represented by a bar graph: The empirical distribution function Φ(x) : R [0, 1] is defined by Φ(x) := the proportion of the range of X x. Thus Φ is monotone increasing with Φ( ) = 0, Φ(+ ) = 1, Φ(x k ) = Φ k between the values of the range of X Φ(x) is constant. The function Φ(x) can be deduced from the above bar graph of X. 2

3 The Stem and Leaf diagram, introduced by J.W. Tukey (1977), gives more information than a barplot. The decimal point is at the If the variable X is continuous rather than discrete, classes are prescribed so that it can be treated as if it where discrete. The number of classes, and the bounds on the classes are left to the software package, or chosen to make the graphics look good. A rule of thumb is to have n classes. The data can then be presented in a table of the form: [k, [b k 1; b k [, n k ] k=1,k such that K n k = n k=1 We define the amplitude of the class as a k := b k b k 1. To simplify, we will only consider constant amplitudes. Rather than a barplot representing the frequencies of X, we use a histogram. The height of the classe in the histogram usually corresponds to the number in the class. The frequency of the class can also be used. 3

4 The definition of the empirical distribution function does not change. One can think of the empirical distribution function as the integral of the histogram of X (using frequencies) just as a distribution function is the integral of the probability density function in probability. 1.2 Two variables Suppose that X : Ω ω 1 and Y : Ω ω 2, that is, using the equivalence Ω 1, n we have couples (x i, y i ) n. Let [X 1,..., X j,..., X J ] and [Y 1,..., Y j,..., Y J ] designate the states of X and Y respectively (quantitative or nominal). Let n jk be the number of elements with state X j and state Y k. 4

5 X Y Y 1 Y k Y K X 1 n 11 n 1k n 1K n x j n j1 n jk n jk n j x J n J1 n Jk n JK n J. n.1 n.k n.k n Table 1: Contingency table Where n j. = k n jk, n.k = j n jk, and n = j n j. = k n.k. The contingency table can be normalised to a frequency table by dividing each number by n to obtain: f jk = n jk n. The conditional frequencies are used to establish the independence of the variable X and Y. Conditional row profile for row j : n jk n j. Conditional column profile for column k : n jk n.k. 1.3 Summary statistics of a quantitative variable X Measures of the center Definition 2 The mode:= the class or center of the class with the highest frequency. The median : Q 2 is defined by Φ(Q 2 ).5 and for all x < Q 2, Φ(x) <.5. The arithmetic mean or average: x def = 1 n n x i. The geometric mean: GM def = ( n x i) 1/n. Note that if y i = x i + a, then ȳ = x + a Measures of dispersion The range: R = x max x min. The interquartile range: IQR def = Q 3 Q 1. The variance: VarX = 1 n (x i x) 2 n ( n ) = 1 x 2 i x 2. n 5

6 To estimate the variance an unbiased estimator is used: s 2 X = 1 n 1 n (x i x) 2. The kth moment: µ k := 1 n n (x i x) k. Standard deviation: σ X = Var X = 1 n n (x i x) 2. The coefficient of variation: c X = σ X x. etc. Note that Var(aX + b) = a 2 Var X. 2 Graphics Graphics should clarify thus be readable. This means a graph should be titled and scales should be clear Nominative variables Column graph Bar graph Pie chart Discrete quantitative variable Bar graph. Empirical distribution function. 2.1 Continuous quantitative variable Histogramme Empirical distribution function Two quantitative variables Scatterplot. 6

7 2.2 Link between two variables The scalar product of two vectors X and Y in R n is defined as (X, Y ) = n x iy i. Note that X 2 = (X, X). We also have (X, Y ) = X Y cos θ where θ is the angle between X, and Y Two quantitative variables X et Y The data (i, x(i), y(i)) n can be represented as vectors X and Y Rn. def We also define X = x(1,..., 1). We have Var X = 1 n (X X, X X) = 1 n X X 2. We compute The covariance: Cov (X, Y ) = 1 n n (x i x)(y i ȳ) = [ 1 n n x ] iy i xȳ = 1 n (X X, Y Ȳ ). Property: Var (X + Y ) = Var X+Var Y + 2Cov(X, Y ). The Bravais Pearson correlation coefficient. r(x, Y ) = Cov(X, Y ) σ X σ Y = cos θ. Where θ is the angle between Y and Ŷ (see below). R 2 = Ŷ Ȳ 2 / Y Ȳ 2 = cos 2 θ, Where θ is as above. Linear regression or least squares regression consists of projecting Y Ȳ onto the span of X X. In other words we solve the problem: min Y Ȳ a(x X) 2. (3) a R to obtain the projection of Y Ȳ onto the space spanned by X X. The projection â(x X) is called Ŷ Ȳ. Equivalently we suppose y i = f(x i ) = ax i + b + ɛ i and compute â and ˆb such that n ɛ2 i = n (âx i + ˆb) 2 is minimal. One obtains Remark 1 The equation Cov (X,Y ) Var X â = b = ȳ â x. Ŷ = â(x x) + ȳ Defines the linear regression line which goes through the point ( x, ȳ). From the contingency table one can calculate: 7

8 The mean: x := 1 n I n i.x i. La variance: σ 2 (X) := 1 n I n i.(x i x) 2. The conditional means : The conditional variances: x j := 1 n.j σ 2 j (X) := 1 n.j I n ij x i. I n ij (x i x j ) 2. The variance of the conditional means of X or explained variance: σ 2 ( x j ) := 1 n J n.j ( x j x) 2. The average conditional variance of X or residual variance: σ 2 j (x) := 1 n j=1 J n.j σj 2 (X). j=1 The correlation quotients: η 2 X/Y and η Y/X: η 2 X/Y : = σ2 ( x j ) σ 2 (X) η 2 Y/X : = σ2 (ȳ i ) σ 2 (Y ) The correlation quotients are used to test for a functional relationship (not necessarily linear) between X and Y. 8

9 2.2.2 Time Series Indices: Elementary indices: I n 0 = P n/p 0. Properties: circularity: I n 0 = I n n I n 0 reversibility: I 0 n = 1/I n 0. Chains: I n 0 = I n n 1 I n 1 n 2 I 1. 0 Synthetic indices: Let {B i } N, be N goods, with respective prices: {p i 0 }N at t = t 0 and {p i n} N at t = t n bought with respective quantities {q0 i } at t = t 0 and {qn} i at t = t n. Simple average index: I n 0 = N pi n N pi 0 Average of indices: The Laspeyre index: I n 0 = 1 N N I i n 0 L n 0 := N pi nq i 0 N pi 0 qi 0 = i p i n ρ i p i. 0 The Paasche index: P n 0 := N pi nq i n N pi 0 qi n = 1. i ρ i pi 0 p i n A simple time series is a data set of the form: (t, x t ) T t=1. Let T t := the trend (4) s t := seasonal variations (5) c t := cyclical variations (6) ɛ t := random variations. (7) We consider the model: x t = T t + s t + c t + ɛ t. 9

10 In certain cases x t = T t s t c t ɛ t In which case we consider Log x t to reduce to the first case. We suppose s t to be periodic with 0 average, that is, there is a k such that The moving averages k : s t = s t+k t k s t+i = 0 t. i=0 for k = 2p + 1, for k = 2p. Forecasting: y t := x t p + x t p x t + + x t+p 2p + 1 y t := x t p/2 + x t p x t+p 1 + x t+p /2 2p Step 1: linear regression of y t as a function of t, ŷ t = ât + ˆb. (8) (9) Step 2: estimate seasonal values, ŝ t = x t ˆT t = x t ŷ t Step 3: because we have estimated s t several times, we take the average Thus ŝ t := 1 I I ŝ t+ik. ˆx t = ŷ t + ŝ t Two nominal variables From the contingency table (cf 1.4) we may test whether two variables X and Y are independent or not with the χ 2 test of Independence s: χ 2 = L l=1 c=1 C (n lc E lc ) 2 where n lc := the observed number E lc := n l.n. c n. Note that E lc is the expected number if the variables are independent. 10 E lc

11 Remark 2 1. If X and Y have no interdependence, then one expects that χ 2 = χ 2 depends on n, L, and C and lim n χ 2 is a known distribution. If C + L 2, we have the following bounds on χ 2 : 0 χ 2 n[min(l, C) 1]. For large n, we approximate χ 2 with the limit distribution to test the hypothesis: H 0 : X and Y are independent. H 0 is rejected if the χ 2 statistic is sufficiently large. More precisely, we fix our tolerance of error, usually at either 5% or 1%. A bound is computed such that the probability of χ 2 exceeding the bound under H 0 is our tolerated error. If this bound is exceeded by our computed statistic H 0 is rejected. 11