Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Chap 3. The Average and the Standard Deviation 1.The Center and the Spread 2.The Average 3.The mean, median and mode 4.The Root-Mean-Square 5.The Standard Deviation 6.The Degrees of Freedom 7.The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/27 spread center  A histogram can be used to summarize large amounts of data giving the center of the histogram and the spread around the center. - center : mean, median - spread : deviation, interquartile range Center and Spread(I) 1. The Center and the Spread

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/27 1. The Center and the Spread  The histogram can be summarized by the center and the spread, but things do not always work out so well.  Reporting only the center and spread of this histogram would miss the two peaks Center and Spread(II)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/27  The average of a list of numbers equals their sum, divided by how many there are. EX) data : 9, 1, 2, 2, 0 sum : = 14 number of data : 5 average : 14/5 = 2.8 Average 2. The Average

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/27  The average is important but not sufficient.  The distribution with the same average varies if the variances are different. average Average and the variance 2. The Average average

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/27 Age-specific average heights and weights Does the average height of people decrease after age 20, dropping about 10cm in 60years? Does the average weight of people decrease after age 20, dropping about 15kg in 60years? 2. The Average

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/27 2. The Average Lexis diagram Cross-sectional data Time-series data Longitudinal data Examine O and X along each 45 degree line. At 1978 O : age 70, 157cm X : age 21, 162cm You need to know the type of data first!

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/ Weight (kg)  The left and the right of the mean are not always same.  It is the median of a histogram that has the value with half the area to the left and half to the right. Histogram for the weight 3. The Mean, Median and Mode

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/27 If the histogram is symmetric around a median, it equals the average. Histogram for the list 1,2,2,3 0% 25% 50% The Mean, Median and Mode

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/27 list 1,2,2,3 0% 50% list 1,2,2,5 0% 50% list 1,2,2,7 0% 50% The average and the histogram The histograms balance when supported at the average 3. The Mean, Median and Mode

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/27 Average is about the same as median Average is bigger than median Average is smaller than median Long right hand tail (right-skewed) Long left hand tail (left-skewed) symmetric Half the area under the histogram lies to the left of the median and half to the right. The tails of a histogram 3. The Mean, Median and Mode

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/27  mode: the most frequent value T hink of ‘ fashion ’. Fashion Mode! 3. The Mean, Median and Mode Histogram is highest at the mode. Histogram and the mode

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/27 4. The Root-Mean-Square  Root-Mean-Square how to do the arithmetic, read it backwards: (1) SQUARE all the entries, getting rid of the signs. (2) Take the MEAN of the squares. (3) Take the square ROOT of the mean. EX) For the list 0, 5, -8, 7, -3 find the RMS Root-mean-square

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/27  SD is a sort of average deviation deviation from the average = each value – average SD  RMS of deviations from the average Ex) for the list 20, 10, 15, 15 find the SD. average = ( )/4 = 15 each deviation from the average : 5, -5, 0, 0 SD: Calculating SD 5. The Standard Deviation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/27  SD is often helpful to think of the way a list of numbers spreads out around the average. Roughly 68% of the entries on a list are within 1 SD of the average. And roughly 95% are within 2 SDs of the average Meaning of SD mean +2SD mean-2SD 95% mean +1SD mean-1SD 68% 5. The Standard Deviation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/27 5. The Standard Deviation Histogram for the heights of women (i)(ii) The heights of the 5,257 women in the Health and Nutrition Examination Survey sample. The region within 1SD of the average is shaded in (i) and 2SD in (ii) 68 – 95 rule

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/27 6. The Degrees of Freedom  The degrees of freedom is as same as the number of actual independent values among all the values.  When calculating SD, we get the average by dividing the sum of the squares of the deviations by “ number of data – 1 ” and then get the root squares of the average.  Why do we use “ number of data – 1 ” instead of “ number of data ” ? The sum of the deviation has to be 0, so they cannot all vary freely. The constraint that the sum equals 0 eliminates one degree of freedom. DF when calculating SD = number of data - 1 Definition of DF

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 24/27 INDEX 1 The Center and the Spread 2 The Average 3 The mean, median and mode 4 The Root-Mean-Square 5 The Standard Deviation 6 The Degrees of freedom 7 The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 25/27  Measurement Error No matter how carefully it was made, a measurement could have come out a bit differently than it did. There exists difference between the exact value and the measurement. Such a difference is a measurement error. Measurement error makes an individual measurement deviate from the exact value. Definition Measurement error = individual measurement – exact value Individual measurement = exact value + measurement error 7. The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 26/27  The SD of a series of repeated measurements becomes SD of the measurement error.  The SD is an estimate of the likely size of the measurement error in a single measurement. Size of the measurement error The size of the measurement error is obtained by SD of individual measurements Individual measurement = exact value + measurement error SD of individual measurement = SD of measurement error 7. The Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 27/27  Bias is the systematic error. Ex) A butcher weighs a steak with his thumb on the scale. Ex) A draper store uses a cloth tape measure shrunk. Bias affects all measurements the same way, pushing them in the same direction. When bias is present, the long-run average of repeated measurements will itself be either too high or too low. The error should not cancel out. Bias Individual measurement = exact value + bias + chance error 7. The Measurement Error