Preliminary Statistics course. Lecture 1: Descriptive Statistics

Transcription

1 Preliminary Statistics course Lecture 1: Descriptive Statistics Rory Macqueen September 2015

2 Organisational Sessions: Sep , V Sep , V Sep. Revision day 25 Sep. Examination Homework: DO in advance! Will be discussed in the following day s lecture Materials: Website prelimsoas.webs.com Moodle Questions: Ask other students via Moodle discussion forum If nobody can help use (rm43@soas.ac.uk) 2

3 Outline Basic Concepts Summation Operator (digression) Descriptive Statistics Numeric Summaries/Summary Statistics Central Tendency Dispersion Shape Covariance/correlation Standardised Data Graphical Techniques M. Barrow Ch.1 provides a good presentation of numerical and graphical descriptive statistics 3

4 Basic Concepts Origin Of Statistics: The collection of information on the population by the state Definition Of Statistics: An arithmetic measure derived from a sample set of data Commonly used as an estimate of a population parameter Functions: Description Inference Prediction 4

5 Basic Concepts Functions of Statistics Descriptive Statistics Statistical Inference A set of methods to describe data A set of methods that use information from a sample to infer something about a population Numerical techniques Graphical techniques 5

6 Basic Concepts Functions of Statistics Time Series The same data is collected repeatedly over a number of time periods (e.g.: UK annual inflation) y t, t = 1, 2,, T Cross-sectional Data is collected from the elements of the sample at one point in time (e.g.: Living Standards Measurement Survey SA 1993) y i, i = 1, 2,, N Panel/Longitudinal The same data is collected from the same elements over a period of time (e.g.: British Household Panel Survey) y it, i = 1, 2,,T; t = 1, 2,,T 6

7 Basic Concepts Level of Measurement Characteristic Nominal Ordinal Interval Ratio Example Gender Preferences Temperature Length, age Distinctiveness Ordered by size Equal intervals Absolute zero Level of measurement of a variable is a classification to describe the nature of information contained within numbers assigned to objects and hence within the variable. 7

8 Basic Concepts Quality of Measurement Instruments Reliability A measurement instrument is reliable if in repeated trials it presents the same measure. The measure may be wrong, but it is the same each time. Eg.: 2003 GDP 2nd quarter growth underestimated due to incorrect construction figures. Validity A measurement instrument is valid if it measures the concept that is intended. Eg.: 2001 Employment Survey found an extra 750,000 workers. (S. Briscoe, 2006, (FT Sep 2006) 8

9 Summation Operator N i=1 X i T t=1 X t 9

10 Summation Operator 5 i=1 X i = X 1 + X 2 + X 3 + X 4 + X 5 Index X i 1 17, , , , ,000 SUM 107,000 10

11 X i Y i X i 2 X i 2 N i=1 P j=1 Summation Operator Common Expressions Multiply the matched pairs X and Y, then sum Square each value of X then sum Sum the values of X then square X i Y j Double Summation 11

12 Rule 1 The sum of a constant: Summation Operator Summation Rules n i=1 a = a + + a = na Rule 2 The sum of a constant times a variable: n i=1 ax i = ax 1 + ax ax n n = a X 1 + X X n = a X i i=1 12

13 Rule 3 Summation Operator Summation Rules Summation is commutative over addition (but not over multiplication): n i=1 (X i +Y i ) = (X 1 +Y 1 ) + (X 2 +Y 2 ) + = X 1 + X X n + Y 1 + Y Y n n n = X i i=1 + Y i i=1 13

14 Rule 3 Summation Operator Summation Rules Summation is commutative over addition (but not over multiplication): n n n (X i Y i ) X i Y i i=1 i=1 i=1 14

15 Numeric Summaries Central Tendencies Mean, Median, Mode Dispersion Variance, Standard Deviation, Range, Percentiles, Inter- Quartile Range Shape Skewness, Kurtosis Measure of Association Covariance, Correlation Standardised Data 15

16 Numeric Summaries Central Tendencies: The Mean Arithmetic mean i = 1, 2, n x = Grouped data C = number of classes x = f k = number of obs. in group k x i n C k=1 C k=1 f k f k x k 16

17 Other means Numeric Summaries Central Tendencies: The Mean Weighted mean x w = Geometric mean n n i=1 w i X i xg = X 1 X 2 X 3 X n 17

18 Numeric Summaries Central Tendencies: The Median Definition The Median The median is the middle value of the distribution (50 th percentile) i.e. 50 percent of the values of the variable are below and above the median respectively. Calculation 1. Put the observations in an ascending/descending order, 2. Find the midpoint observation, 3. Location of Median: (n+1)/2. Quartiles If odd sample: Median is the value of the middle observation. If even sample: Median is the average value of the two middle observations. Quartiles are found by dividing the distribution into four parts (same method as for median). 18

19 Numeric Summaries Central Tendencies: The Mode The Mode Is the most frequently occurring value among the entire sample. 19

20 Numeric Summaries Central Tendencies relative strengths Mean: Interval/ratio data Sensitive to outliers Useful in further statistics A reasonable measure for symmetrically distributed variables Median Ordinal, interval, and ratio variables Robust in terms of the shape of the distribution and outliers Mode: Nominal, ordinal, interval, and ratio A dataset can be bi- or multi-modal 20

21 Range Numeric Summaries Measures of Dispersion Simply the spread of values of the data set Measured by: Maximum Minimum value Only uses two values i.e. discards potentially a lot of information Interquartile Range Difference between the third and the first quartile Measured by: Upper Quartile Lower Quartile 21

22 Variance Numeric Summaries Measures of Dispersion The variance is the average of all squared deviations from the mean: σ 2 = (X i X) 2 n With sample size n Unbiased estimator for the variance: s 2 = (X i X) 2 n 1 22

23 Numeric Summaries Measures of Dispersion Standard Deviation The standard deviation is given by the square root of the variance: σ = (X i X) 2 n and s = (X i X) 2 n 1 with n-1 degrees of freedom* The standard deviation is measured in the same units as the data. *degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. 23

24 Numeric Summaries Measures of Dispersion Coefficient of Variation Measure of relative dispersion Provides a method of comparing the variation of variables measured in different units Expresses the standard deviation as a proportion of the mean σ μ (If for example 0.81, the s.d. is about 80% of the mean) 24

25 Skewness Numeric Summaries Measures of Shape Show how asymmetric a distribution is Coefficient of skewness: 1 ( x i x ) 3 n s Distributions can be: Zero symmetric Positively skewed, skewed to the right (long right tail) Negatively skewed, skewed to the left (long left tail) 25

26 Kurtosis Numeric Summaries Measures of Shape Measure of peakedness Coefficient of excess kurtosis: 1 ( x i x ) 4 3 n s Distributions can be: Mesokurtic (eg. Normal distribution) Leptokurtic (sharp peak and slim tails) Platykurtic (flat and fat tails) 26

27 Covariance Numeric Summaries Measure of Association A measure of how two variables vary together If both variables move in the same direction the covariance will be positive If the variables move in different directions the covariance will be negative The problem is that there is no upper limit; the value of the covariance depends on the units of measurement cov x, y = (x i x )(y i y) n 27

28 Numeric Summaries Measure of Association Correlation Equivalent to the covariance for standardised variables Range: -1 ρ 1 0 = no correlation 1 = a perfect linear positive correlation -1 = a perfect linear negative correlation Unit free ρ = cov(x, y) σ(x)σ(y) 28

29 Numeric Summaries Standardised Data Useful transformation of data: Subtract the mean and divide by an estimate of the standard deviation: z i = x i x s New variable is called standardised (or z- score) 29

30 Numeric Summaries Standardised Data Male Female Mean 19,500 16,800 SD 4,750 3,800 Salary 31,375 26,800 Above mean 11,875 10,000 A man and a woman are arguing about their career records. The man says he earns more that her, and hence is more successful. The woman argues that women are discriminated against and she, relative to women, is doing better than him, relative to men. Who is right? The man receives more salary above the mean compared to the woman. But women salaries are less dispersed than men s. The z-scores give the salary of each in terms of SDs from their mean Man z-score = 2.52, Woman z-score = 2.63 i.e. The man is 2.5 SDs above the male mean salary and the woman is 2.63 SDs above the women mean salary. She is nearer the top of the female distribution than is the man. 30

31 Numeric Summaries Moments about the Mean r th moment 1st moment (=0) 2nd moment (variance) 3rd moment (used for skewness) 4th moment (used for kurtosis) m r = m 1 = m 2 = m 3 = m 4 = (X i μ X ) r N (X i μ X ) 1 N (X i μ X ) 2 N (X i μ X ) 3 N (X i μ X ) 4 N 31

32 Graphical Techniques Time-series data Line Graph Evolution of a variable over time Informative about trends, seasonal patterns, cycles, etc Histogram For both time-series and cross-section data Proportion (or frequency) of observations falling in different classes / bins Informative about the shape of the distribution Scatter Diagram (or XY plot) Informative about the relationship between two variables Complement to the correlation coefficient 32

33 Graphical Techniques Time-series data E.g.: cocoa futures prices, correlation with non-commercial traders long positions, and returns 4000 ICE Line Graph ICE ncom_long Scatter Diagram Histogram Density return

34 Bar Chart Graphical Techniques Cross-section data Shows number (frequency) of observations falling in each category 34

35 Graphical Techniques Cross-section data Histogram Similar to bar chart (sometimes with different sized bins) Nominal or ordinal? What s missing from this graph? 35

36 Box Plot Graphical Techniques Cross-section data Shows the min/max, median, and quartiles on a single diagram 36