University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

81

Pages:

29
Author:

SacteraniLiarz2t

The goal of statistics is to Report data in meaningful ways Make predictions about future events 40 Course 15 Course 6 # of people 30 20 10 0 150 160 170 180 190 200 Weight Statistics has 3+ components Data analysis Descriptive statistics Probability calculations Statistical inference Inferential statistics Models .... Describing a state – Descriptive statistics – Capturing a picture of the data) – This was the origin of statistics – Started for gambling First some descriptive statistics 15.301 is the “best class ever”? 100 15 75 5 1 6 50 25 0 Bad Good Very good Excelent 12% 13% 34% 41% Central tendencies • Representing central tendencies of distributions is a very efficient way to understand something about it. • Mode • Median • Mean The Mode • The most “popular” frequent occurring instance in the sample. – This is the only central tendency that can be used with a nominal scale • The mode is sensitive to aggregation of categories – Age 18 vs age 18-21 • Sometimes there are multiple modes – Bimodal distributions The Median • The median is a value which 1/2 of the values are above it and 1/2 below • After sorting the values by magnitude, the mode is at the (n+1)/2 location • 123, 85, 34, 20, 18, 15, 14 ‹ 20 • 123, 85, 34, 20, 18, 15 ‹ (20 + 34)/2 = 27 • When data is grouped, calculating the mode is a bit more complex The Mean • • • • Mean = (∑Xi) /n The most important statistic Used for many other computations Stable – Smallest mean square deviations from it • Sensitive to extreme values • Not “well behaved’ in non-standard distributions Location of central tendencies Normal Mean Mode Median 1.0 .8 .6 .4 .2 0 Location of central tendencies Bimodal Mean Mode Median 1.0 .8 .6 .4 .2 0 Location of central tendencies Skew to right Mean Mode Median .5 .4 .3 .2 .1 0 Location of central tendencies Skew to left Mean Mode Median .5 .4 .3 .2 .1 0 Distribution descriptors • The Range – The range is (Max - Min) • Interquartile range – Calculating is similar to median – Q3-Q1 (1/2 of the observations) Variation I • Variance (s^2) – ∑(Xi-m)^2/(n) – ∑(Xi-m)^2/(n-1) • Standard deviation (s) – Square root of variance – Standard deviation is in the same units as the distribution Variation II • Variance (s ^2) is: • insensitive to transformations consisting of adding a constant. • sensitive to transformations consisting of multiplying by a constant. Describing scores: Z scores z = (r-m) / s m =0,s=1 T scores m = 50 , s = 10 SAT, GRE etc. Confidence in estimates? • How sure can we be that we know the mean of the distribution, for example? • Standard error of the mean – m^2 / Square root of N The Correlation • The relationship between 2 variables does not have to be linear – But in many cases they are • Positive and negative correlations Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? Estimating correlations in scatter grams • What is the correlation here? The correlations were: • • • • • • • 1 ¯ 0.1 2 ¯ 0.3 3 ¯ 0.5 4 ¯ 0.7 5 ¯ 0.9 6 ¯ 0.99 7 ¯ 01 What is a correlation? • What line to pick? – Sum of all deviations from the line is 0 – The sum of square deviations of the points from the line is minimal. • R = Sxy / Sx * Sy – The relationship of their joint standard deviation to their individual standard deviation • R^2 is the amount of explained variance Summary One of the main usages of statistics is to describe data Central tendencies: Mean, Mode, Median Distribution tendencies: Variance, IQR, Correlations