AMS 5 NUMERICAL DESCRIPTIVE METHODS

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1 AMS 5 NUMERICAL DESCRIPTIVE METHODS

2 Introduction A histogram provides a graphical description of the distribution of a sample of data. If we want to summarize the properties of such a distribution we can measure the center and the spread of the histogram.

3 Introduction These two histograms correspond to samples with the same center. The spread of the sample on top is smaller than that of the sample in the bottom

4 The Average Table: Maximum daily ozone values (in PSI in L.A. on 120 typical days in 1989)

5 The Average Frequency Maximum Daily Ozone (PSI)

6 The Average What was the typical ozone level in 1989? How should we define what we mean by a typical number? The most frequently used definition, is the simple rule; add the numbers up and divide them by how many there are. The average (or arithmetic mean) ) of a list of numbers equals their sum divided by how many there are. 1 n = = = i n i = 1 x x 10,729 / PSI

7 The Median Frequency median average Maximum Daily Ozone (PSI) We can easily observe that only 42 out of the 120 observations are above the average (i.e. only 35%). This is because the histogram is not symmetric. A Histogram balances when supported at the Average. The median is the value with half the area of the histogram to the left and half to the right.

8 The Median A symmetric histogram will look like this. In this case 50% of the data are above the average, i.e. the median and the average coincide.

9 Calculating the Median Sort your data. If you have an odd number of observations your median is exactly the middle number of the ordered ones (i.e. with 9 ordered numbers, the median is the 5 th ordered observation). If you have an even number of observations, like in our case with the ozone example where n=120, the there are two numbers that could be the middle, the 60 th and the 61 st ordered observations (with values 79 and 80 PSI respectively). In order to resolve this conflict we can choose as the median the average of this two numbers, and therefore the median in our example is (79+80)/2=79.5.

10 Average or Median

11 The Root-Mean Mean-Square Back in the ozone example, and we saw that the typical ozone level is either around 80 or 90, depending on how to define typical. But give or take how much? In other words what is the typical amount by which the numbers differ or deviate from the middle one? What we need is a numerical measure of the spread of the data. Before defining this measure of size lets, answer to an even simpler question. How big are our numbers? To make things easier lets suppose that we only had the following 5 numbers: 0, 5, -8, 7, -3 The average is ( )/5 = 0.2 but this is a very poor measure of size, since it allows the positives to be cancelled with the negatives.

12 The Root-Mean Mean-Square The simpler way around this problem would be to wipe out the signs by averaging the absolute values. The average neglecting signs then is ( )/5 = 4.6 and it gives us a measure of size. Another measure is the root-mean mean-square SQUARE all the entries Take the MEAN of the squares Take the square ROOT of the mean. Therefore : ( 8) ( 3) r.m.s. size= = 5.4 5

13 The Standard Deviation Back to our initial question. What is the typical amount by which the numbers differ or deviate from the middle one? By choosing as a middle number the average, what we really want to calculate is a measure of size of every entry from the average. We will use the root mean square as a measure of size and the final measure of spread that we will obtain will call it standard deviation (SD).

14 The Standard Deviation Therefore SD = r.m.s. deviation from the average = r.m.s. (entry average) To calculate the standard deviation of a sample follow the steps: Calculate the average Calculate the list of deviations from the average by taking the difference between each entry and the average. Calculated the r.m.s. size of the resulting list.

15 The Standard Deviation Example: Lets go back to the simple example with only the following 5 numbers: 0, 5, -8, 7, The average is ( )/5 = In order to find the deviations from the average we just have to subtract the average from each entry: -0.2, 4.8, -8.2, 6.8, Find the r.m.s. size of the deviations: ( 0.2) ( 8.2) ( 3.2) SD= =

16 The Standard Deviation Shortcut SD Formula n n 2 xi i= 1 n i= 1 For the ozone example: = 1 n Observation number (i) x i x 2 i ( /120) 2 SD= = sum x i 2

17 The Standard Deviation The SD comes out in the same units as the data. Do not confuse the SD of a list with its r.m.s. size. The SD is the r.m.s. not of the original numbers on the list, but of their deviations from the average. In many data sets (especially when the distribution is symmetric) the following rule of thumb applies: 1. Roughly 68% of the observations are within one SD of the average. 2. Roughly 95% of the observations are within two SDs of the average. 3. Roughly 99% of the observations are within three SDs of the average.

18 Using a Statistical Calculator

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 5: Summary Statistics Tessa L. Childers-Day UC Berkeley 30 June 2014 By the end of this lecture... You will be able to: Describe a data set by its: