¡Descarga Unidad 3 (inglés) y más Apuntes en PDF de Idioma Inglés solo en Docsity!
Percentiles
Summarizing univariate data (I):
Measures of location
Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
Measures of location
Percentiles
Introduction
Measures of central tendency
Other measures of location
Measures of location
Intro PercentilesCentral
Introduction: use of descriptive measures
Consider the following data about the age of 20 students in a classroom: 20 19 21 20 19 19 19 20 21 25 20 24 20 21 23 21 20 19 20 21
We want to summarize them by means of several descriptive measures. We aim to answer the following questions: I (^) What is the average age? I (^) What is the maximum age of the 75% youngest people in the sample? I (^) How much the ages are spread out? Are most of the values close to each other? What is the biggest difference between values?
Intro PercentilesCentral
Types of descriptive measures
I (^) Measures of location.- They determine certain positions in the sample. I (^) Measures of central tendency.- They provide us with some idea of centre or middle of a set of data. Averages. Example: What is the average age? I (^) Other measures of location.- They determine other positions in the sample, different from central positions. Example: What is the maximum age of the 75% youngest people in the sample? I (^) Measures of spread or variability.- They inform us about how widely the values of the variable are spread out or dispersed. Example: How much the ages are spread out?
Percentiles
Measures of central tendency
I (^) Arithmetic mean I (^) Weighted mean I (^) Geometric mean I (^) Median I (^) Mode
Measures of location
Percentiles
Arithmetic mean
I (^) Definition: Sum of the n observations divided by n. I (^) Mathematical formula: If X takes the values x 1 ,... , xk with frequencies n 1 ,... , nk , then:
x =
x 1 n 1 +... + xn nk n
I (^) Example: 20 19 21 20 19 19 19 20 21 25 20 24 20 21 23 21 20 19 20 21
x =
Measures of location
Intro PercentilesCentral
Properties of arithmetic mean
P1 Positive differences wrt the arithmetic mean exactly cancel out the negative differences:
(x 1 − x)n 1 +... + (xk − x)nk = 0.
P2 If Y = cX + d then, y = cx + d. (You do not need to construct the frequency table of Y ). P3 Suppose that the sample is divided into r groups (sub-samples) of known sizes (n = n 1 +... + nr ). You know the arithmetic mean in each group. What is the mean in the whole sample? I (^) Denote xi the arithmetic mean of the i-th group. I (^) Then x = x^1 ·^ n^1 +... n +xk^ ·^ nk.
Intro PercentilesCentral
Weighted mean
I (^) Mathematical formula: Given r numbers y 1 ,... , yr with respective weights w 1 ,... , wr : yw = y^1 w^ w 11 ++......+yrw^ wrr I (^) Example: I (^) Goal: Estimating fuel costs during a vacation drive. I (^) Solution: Calculation of weighted mean. Values: price per gallon; Weight: purchased gasoline in gallons. yi (price per gallon) wi (Qty of gas (gallons) 1.12 29 1.18 31 1.20 13 1.25 13
y (^) w =
- 12 · 29 + 1. 18 · 31 + 1. 20 · 13 + 1. 25 · 13 29 + 31 + 13 + 13 ≈^1.^17
Percentiles
Example of calculation of the median: odd sample size
Dataset: 1, 5, 1, 8, 7. I (^) Calculation from raw data. I (^) Start by sorting the values: 1, 1, 5, 7, 8. I (^) In this case, the median is 5, since it is the middle observation in the ordered list. I (^) Calculation from frequency table: xi ni pi Pi 1 2 40% 40% 5 1 20% 60% 7 1 20% 80% 8 1 20% 100% The median is x 2 = 5, since P 1 = 40% < 50% < P 2 = 60%.
Measures of location
Percentiles
Example of calculation of the median: even sample size
Dataset: 1, 5, 2, 8, 7, 2. I (^) Calculation from raw data. I (^) Start by sorting the values: 1, 2, 2, 5, 7, 8. I (^) In this case, the half sum of the two middlemost terms is (2 + 5)/2 = 3.5. Therefore, the median is 3.5 since it is the average of the middle observations in the ordered list. I (^) Calculation from frequency table: xi ni pi Pi 1 1 16.66% 16.66% 2 2 33.33% 50% 5 1 16.66% 66.66% 7 1 16.66% 83.33% 8 1 16.66% 100% The median is x^2 + 2 x^3 = 3. 5 , since P 2 = 50%.
Measures of location
Intro PercentilesCentral
Mode
I (^) Definition: The most frequent value in the sample. I (^) Mathematical calculation: Choose the value(s) with maximum frequency. I (^) Examples: I (^) Dataset: 1, 2, 1, 4, 2, 2, 4, 4, 4, 5. The mode is number 4. I (^) Dataset: 1, 2, 1, 4, 2, 2, 4, 4, 5. There are two modes: number 2 and number 4.
Intro PercentilesCentral
Other measures of location: percentiles
I (^) Definition: The value of the variable below which a certain percent of observations fall^2. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found. I (^) Mathematical calculation using cumulative percentages. There are two possible situations: a) There exists i with Pi < k% < Pi+1. Then the kth percentile is xi+1. b) There exists i with Pi = k%. Then the kth percentile is xi^ + 2 x i+1.
(^2) This is an informal definition. The formal one is out of the scope of this course.
Percentiles
Example of calculation of percentiles
Consider the following frequency table:
xi ni pi Pi 1 2 40% 40% 5 1 20% 60% 7 1 20% 80% 8 1 20% 100%
I (^) The 70% percentile x 3 = 7 since P 2 = 60% < 70% < P 3 = 80%. I (^) The 80% percentile is x^3 + 2 x^4 = 7.5 since P 3 = 80%.
Percentiles
Property of percentiles (including the median)
I (^) If Y = cX + d with c > 0, then, the kth percentile of Y coincides with cperck + d, where perck is the kth percentile of X. (You do not need to construct the frequency table of Y ). I (^) Example: Consider a sample of civil servants. Let X be their The median current salary of civil servants is 1300 euros. All salaries are being cut by 5 percent next year. Let Y the income after the cut. Y = (1 − 0 .05)X = 0. 95 X. The the new median salary of civil servants (median of Y ) will be 1300 · 0 .95 = 1235.
