CS110 Personal Computing 1

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1 CS110 Personal Computing FORMULAS Understanding formulas The trainings are starting to introduce basic formulas and functions We re also starting to look at the statistical analysis part of the class We noticed about ½ the class struggled with these basic formulas So Percentages If a city has 20% growth per year: How do we calculate the estimated population? By multiplying 20% times the current number And add that to the current number 2013 has 4000 people For 2014, that number is equal to : Calculate the increase (4000 * 20%) = 800 Add that to 4000 = 4800 CS110 Personal Computing 1

2 Let s look at this Why do we use absolute ref? Let s look at the action of copy and paste When a relative address is copied and pasted The references are updated This allows the same formula to be used When an absolute address is copied and pasted The references are not updated This allows constants or parameters to be used Back to our example CS110 Personal Computing 2

3 Copying formula Pasting formula See how the relative references were updated Note the absolute ref It stays the same: CS110 Personal Computing 3

4 Ratios What is a ratio? a ratio is a relationship between two numbers of the same kind Usually expressed A to B or A:B What does it mean? The ratio of bananas to apples is 1:4 The ratio of children to couples is 2.3:1 How do we compute ratios? If we want to know the ratio of cars to people We know there are 400 cars We know there are 200 people What is the ratio? 400:200 Can this be simplified? Yes, by dividing the first by the second Gives us 2:1 What do we use ratios for? To compare two quantities To simplify them to understandable numbers The ratio of children to couples is : 57,500:25,000 or 2.3:1 Which is easier to understand To extrapolate future numbers If the ratio of bananas to apples in a delivery is 4:1 A delivery has 50 apples CS110 Personal Computing 4

5 Averages How do we determine averages? The sum of the values divided by the number of the values So if : Bob as 4 bananas Terri has 2 bananas Sammy has 3 bananas What is the average number of bananas each of them has? What do we use averages for? To extrapolate numbers If each person need an average of 2 liters /day, how much water will 10,000 people need? The more samples there are, the more reliable the average Bob 5 liters Sammy 1 liters Average is 3 liters Small numbers have trouble with outliers What is an outlier? An observation that is numerically distant from the rest of the data What causes outliers? Error in measurement, outside influences Example if my study is tracking time to compute an algorithm, an outlier could be caused by someone else on the computer using the computer cycles or memory CS110 Personal Computing 5

6 How do they affect numbers? They skew the average We record daily water usage With Bob Bob 11 liters (watered his garden as well) Sammy 1.8 liters Tommy 2 liters Andrea 2.2 liters Average is 4.25 liters. If we used this number for 10,000 people, we would have 22,500 liters too many Without Bob Sammy 1.8 liters Tommy 2 liters Andrea 2.2 liters Average is 2 liters If we use this number, we would be right on. How do we get rid of outliers? First we have to identify them We use standard deviation for this Take those values out of the study Re-compute the averages Standard deviation Used to identify the variance in the data The formula for a complete population: If N = population size and Avg = deviation is: 1 N v i N, then the standard i=1 N v i A 2 N CS110 Personal Computing 6

7 How does this apply to outliers? We use the standard deviation to identify the outliers. Let s say we identify anything that is more than 2 times the standard deviation as an outlier Let s go back to our water example: Water example First we compute the average: Bob 11 liters (watered his garden as well) Sammy 1.8 liters Tommy 2 liters Andrea 2.2 liters Jeff 1.9 liters Jackie 2.1 liters Average is 3.5 liters. Water example Next, we compute the sum of the squares Compare to avg Deviation 2 Bob = -7.5 (-7.5) 2 = Sammy = 1.7 (1.7) 2 =2.89 Tommy = 1.5 (1.5) 2 = 2.25 Andrea = 1.3 (1.3) 2 = 1.69 Jeff = 1.6 (1.6) 2 = 2.56 Jackie (1.4) 2 = 1.96 So TotalStandard Deviation = CS110 Personal Computing 7

8 Finishing standard deviation With the sum of the squares = So our standard deviation is 3.35 Identifying the outliers Standard deviation x standard deviation 6.7 Average = 3.5 Throw out anything Over 10.2 Under -3.2 Value with Bob is greater than 10.2 Bob 11 Value Sammy 1.8 Tommy 2 Andrea 2.2 Jeff 1.9 Jackie 2.1 Re-compute the average Value Sammy 1.8 Tommy 2 Andrea 2.2 Jeff 1.9 Jackie 2.1 CS110 Personal Computing 8

9 How to do this in Excel? First, set up standard deviation And average Eliminate the outliers CS110 Personal Computing 9

10 And now compute new average Confidence interval Given in a range and a percentage. Says that with the percentage given, the actual value will fall within the range I.E. given: A normal distribution a 95% confidence interval A range between 9 and 11 Says that at least 95% of the time, the value calculated will fall between 9 and 11. What is a normal distribution? CS110 Personal Computing 10

11 How is this helpful? When you calculate averages, you assign them a confidence interval and a range. This tells your readers how reliable your data is Calculated by: Assumes normal distribution Standard deviation Given a specific confidence level Number of data point Calculate in Excel Improve the interval By increasing the number of samples CS110 Personal Computing 11

12 Improve the interval Decrease the confidence level Median The numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median can be used as a measure of location when a distribution is skewed when end-values are not known when one requires reduced importance to be attached to outliers A disadvantage of the median is the difficulty of handling it theoretically Median Given the following sequence, what would be the median of the values: 1,2,2,3,3,3,4,5,5,6,6,6,14 CS110 Personal Computing 12

13 In Excel Mode In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution Very useful for discreet functions rather than continuous functions If more than one, the series can described as bimodal or multimodal Mode What would be the mode of the series: 1,2,3,3,4,5,6,7,7,7,8,8,9,10,10,11,12 How about: 1,2,3,3,4,5,5,6,6 1,2,3,3,4,5,5,5,6,6,6 CS110 Personal Computing 13

14 In Excel References CS110 Personal Computing 14