University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

270

Pages:

3
Author:

Niklas

LECTURE 7 Exercise 5 6 10 1 1 0 1 3 2 5 0 5 0 2 8 15 50 0 0 1 2 2 3 3 5 5 5 6 8 10 15 50 50 N=14 Q1: 2, 3, 3, 5, 5, 7.5 position 5 Q2: 3.75, 3, 3, 5, 5, 7.5 position 5 Q3: 3, 3, 3, 5, 5, 7.5 position 8.25 IQR = Q3 - Q1 = 8.25 - 1.75 = 6.5 Lw = Q1 - 1.5 * IQR = 1.75 - 1.5 * 6.5 = -8 Uw = Q3 + 1.5 * IQR = 8.25 + 1.5 * 6.5 = 18 Distance from the center A way of measuring dispersion The measurement of the distance of each observation from the center: (y_i - ȳ)^2 We make the mean of such quantities: (1/N) * Σ(y_i - ȳ)^2 The variance of the observation y_1, y_2, ..., y_N The mean of the squares of the deviation (distance) of each observation from the mean. σ^2 = (1/N) * Σ(y_i - ȳ)^2 The variance of the variable Y in symbol: σ_Y^2 or V(Y) These two distributions have the same mean. However, the dataset on the right has a greater variability, and, hence, a higher variance. Exercise 5, 5, 6, 6, 6, 7, 8, 5 x̄ = (5*3 + 6*4 + 7*1 + 8*1)/8 = 6 x_i - x̄: 5-6, 5-6, 6-6, 6-6, 6-6, 7-6, 8-6, 5-6 (x_i - x̄)^2: 1, 1, 0, 0, 0, 1, 4, 1 n_i(x_i - x̄)^2: 3, 3, 0, 0, 0, 1, 4, 1 σ^2 = 8/8 = 1 The standard deviation doesn't sum up to zero. We aren't using frequency. The standard deviation is the square root of the variance. It's expressed in the same units as the variable. It's a better measure of variability. σ = √(σ^2) The sum of squares L represents the sum of the squared deviations of the observations from their mean. L = Σ(y_i - ȳ)^2 Example x_i x_i - x̄ (x_i - x̄)^2 x_i^2 3 -2 4 9 5 0 0 25 7 2 4 49 Σ 15 8 83 Mean: x̄ = 5 Mean of the squares: 83/3 = 27.6 Square of the mean: 25 d² = 27.6 - 25 = 2.6