# Standard deviation V1

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### Standard deviation:

Standard deviation Stimulating Thinking Patterns 3/8/2018 Trademark- TrainFirm 1

### What is Standard deviation?:

What is Standard deviation? Let me try to explain standard deviation using a very simple and common example Example - Time taken to withdraw money from ATM. Let's assume that you have 20 people of ahead of you in queue to withdraw money and you measured the time taken by 10 people ahead of you in minutes. Measured values are …5,6,7,8,5,3,2,4,7,8 We see that values are not consistent and time taken by each customer vary from other customers. With an average of 5.5 minutes (Mean) Is that all you could estimate?? We can see one customer even did it in 2 mins and another person took 8 minutes.. So how do you account for those inconsistencies ? Standard deviation is nothing but the average time any customer is expected to deviate or vary from a standard reference value of time taken to withdraw money . Generally the standard reference value is always considered as the “Mean”. In this example mean of time taken by 10 people ahead of you to withdraw money and the value is 5.5 minutes. 3/13/2018 Trademark- TrainFirm 2 Customer  Time in Mins  1 5  2 6  3 7  4 8  5 5  6 3  7 2  8 4  9 7  10 8 5.5 Average

### How to calculate mean:

How to calculate mean Below are your 10 observations you have timed in minutes.. 5,6,7,8,5,3,2,4,7,8 So the first step is to set that standard reference value which is nothing but the "mean"  How do we calculate the mean.. Simple - Sum of all 10 observations / Total number of observation 55/10 = 5.5 minutes = Sample mean 3/8/2018 Trademark- TrainFirm 3

### How to calculate deviation:

How to calculate deviation Now how do we calculate how much each customer ( 5,6,7,8,5,3,2,4,7,8) deviated from the Mean of 5.5 Let us take the difference of each value from the mean they are .5 , -.5, -1.5,-2.5,.5,2.5,3.5,1.5,-1.5,-2.5 Now, let’s try to find the average value of deviations by applying the same formula for mean.. Will it work?? Unfortunately not.. since there are negative & positive values and mean is the central value, when you take the sum the answer would come as “0” 3/13/2018 Trademark- TrainFirm 4   Time in Mins Average in Mins Deviation   5 5.5 0.5   6 5.5 -0.5   7 5.5 -1.5   8 5.5 -2.5   5 5.5 0.5   3 5.5 2.5   2 5.5 3.5   4 5.5 1.5   7 5.5 -1.5   8 5.5 -2.5 Average 5.5   0

### How to calculate variance and std deviation:

How to calculate variance and std deviation To overcome this challenge let us try to square the differences (.5 , -.5, -1.5,-2.5,.5,2.5,3.5,1.5,-1.5,-2.5) to turn negatives to positives After squaring it becomes .25, .25,2.25,6.25,.25,6.25,12.25,2.25,2,25,6.25 Now apply the same formula for Mean of the squares = sum of all / (n-1)*.. “Hang on there is some change why (n-1) instead of n Simple logic.. since we are dealing with small samples there will always be some amount of error or downward bias in the estimate, to compensate for that bias  and to make it unbiased, we reduce one observation from the denominator hence (n-1) So means of squares = 38.5 / 9 = 4.277 to bring back the mean of squares to the original value let’s take the square root of the mean of squares = sqrt of 4.277  =  2.06 =  std deviation  Job done.. 3/13/2018 Trademark- TrainFirm 5 Time in Mins Average in Mins Deviation Squared value 5 5.5 0.5 0.25 6 5.5 -0.5 0.25 7 5.5 -1.5 2.25 8 5.5 -2.5 6.25 5 5.5 0.5 0.25 3 5.5 2.5 6.25 2 5.5 3.5 12.25 4 5.5 1.5 2.25 7 5.5 -1.5 2.25 8 5.5 -2.5 6.25 SUM 0 38.5

### Let us write down a formula:

Let us write down a formula 3/8/2018 Trademark- TrainFirm 6 Average x = 5.5 minutes Standard Deviation = 2.04 minutes Standard deviation Calculation : A - subtract each data value from the mean B - square each difference C - sum these values D - divide by 1 less than the number of data values E - take the square root of the result  (x - x i ) 2 n-1 n  xi i =1

### How to interpret standard deviation:

How to interpret standard deviation So we conclude that any observation is having an average deviation of 2.06 minutes from the mean of 5.5 Or  in this example any customer would deviate by an average time of 2.06  minutes more or less from the average of. 5.5 to withdraw cash. i.e. either 3.44 mins or 7.56 mins. 3/13/2018 Trademark- TrainFirm 7 Scenario Calculation Your Waiting time Best Case (3.44 mins*10 people ahead ) 34.4 mins (All customers fast) Worst case (7.56 mins*10 people ahead ) 75.6 mins (All customers slow) Most likely (5.5 mins *10 people ahead) 55 mins (Normal distribution) 