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1、.STANDARD DEVIATION*Standard deviation can typically be described as the deviation a piece of date differs from the mean.*Standard deviation can almost be described as the “mean of the mean” it is a form of statistical analysis *You do this when you have data that is distributed normally or Normal d

2、istribution. This means that the pieces of data dont differ from the mean in extremes. *Standard deviation uses past history or data to extrapolate future events. It shows the tendency of the data and how it is spread out. (-Subtracted from the mean-+-added to the mean-)*This is known as a bell curv

3、e. The line through the middle represents the mean or average of the data. Each section is a percent of the data and is one standard deviation away from the meanRed- the red makes up 68% of the entire data; each half is only 34%. (Most of the data falls in this category)Green- each half of the green

4、 is 14% making the green a total of 28% but added to the red makes up for approximately 95% of the data.Blue- Blue represent 2% each but added to the total amount it is 99% (almost all data in this section would be outliers)White- a very small percent often not recognized as part of the data (anythi

5、ng in this section would be an extreme outlier)Fraction Number of Standard Of Data Deviations from Mean 50.0% .674 68.3 1.000 90.0 1.645 95.0 1.960 95.4 2.000 98.0 2.326 99.0 2.576 99.7 3.000*This shows the percent of the data will fall in how many standard deviations*Most of the data is found withi

6、n the first two standard deviations away from the mean negative or positive*The wider and lower bell curves represent a higher standard deviation amount*A steeper more condensed curve represents a smaller standard deviation amount and the data is more clustered around the mean *One form of calculati

7、ng an average is the mean. *The mean is the sum of all the pieces of data divided by the total number of pieces of data Mean = Sum of data Total # pieces of dataEx. 5,6,7,5,6,4,8,9 Sum of data Total # pieces of data508Mean = 6.25*One equation to determine the standard deviation is S= Standard Deviat

8、ionå= The sum of the pieces of dataX= your piece of dataM= meanN= your number of pieces of data*However all this seems very complicating to find of standard deviation when there is an easier way to do it*You will need several pieces of data to determine the mean Ex. 5,7,4,9,5,7,6,4,8,5So now yo

9、u need to calculate the meanMean = Sum of data Total # pieces of data6010Mean= 6Now you need to subtract each piece of data from the mean and then square it· Even if you are subtracting a larger number than the mean and it becomes negative dont worry because when you square a negative it become

10、s positive.#Data pieceMean Subtracted From DataSquared Answer155-6= -11277-6= 11344-6= -24499-6= 39555-6= -11677-6= 11766-6= 00844-6= -24988-6= 241055-6= -11Total= 26*Now you take the total number of squared answers and divide it by the number of pieces of dataTotal Squared Ans# Of pieces of data261

11、0 = 2.6*The total squared answer can also be called the Variance*Now take the square root of the number and you have your Standard deviationStandard Deviation = (of squared total / # of pieces of data)Standard Deviation = (2.6)Standard Deviation = 1.61*There you have your deviation number!*This is a

12、 graph of a bell curve and how it would be labeled after you find the specific standard deviation 1.17-2.78-4.39- 6 -7.61-9.22-10.83*You can determine the percentage error Percentage error=(Standard Deviation / Mean) x 100Percentage error= (1.61 / 6) x 100Percentage error= (.268) x 100Percentage err

13、or= 26.8%*This is a good indicator on whether or not the mean is a good statistic *Standard deviation gives you a basis for interpreting the data in regards to probability*The Variance is the total of the pieces of data subtracted from the mean then squaredVariance= (Data piece Mean) 2*Standard Devi

14、ation is typically used as a component with other indicators to give a real statistical analysis of the given data*Standard Deviation can be used with a variety of real life situations like, stock market analysis, medical concepts, showing the differences in various pieces of data*But typically it i

15、s used to show stock market change and analysis.Ex. Lets take 20 samples of stock market closing to determine the mean and standard deviation of the stock #Piece of dataMean Piece of dataSquared Answer1109.00109.00-112.30=-3.3010.912103.06103.06-112.30=-9.2485.383102.75102.75-112.30=-9.5591.264108.0

16、0108.00-112.30=-4.3018.525107.56107.56-112.30=-4.7422.476105.25105.25-112.30=-7.0549.757107.69107.69-112.30=-4.6221.308108.63108.63-112.30=-3.6813.539107.00107.00-112.30=-5.3028.1210109.00109.00-112.30=-3.3010.9111110.00110.00-112.30=-2.305.3012112.75112.75-112.30=0.450.2013113.50113.50-112.30=1.201

17、.4314114.25114.25-112.30=1.953.7915115.25115.25-112.30=2.958.6816121.50121.50-112.30=9.2084.5817126.88126.88-112.30=14.57212.3418122.50122.50-112.30=10.20103.9719119.00119.00-112.30=6.7044.8520122.50122.50-112.30=10.20103.97Total= 921.28#1. Mean = Sum of data Total # pieces of data2246.0620Mean= 112

18、.30#2. Total Squared Ans # Of pieces of data921.2820=46.064#3. Standard Deviation = (of squared total / # of pieces of data)Standard Deviation = (46.064)Standard Deviation = 6.787Percentage error=(Standard Deviation / Mean) x 100Percentage error = (6.787 / 112.30) x 100Percentage error = (.060400) x

19、 100Percentage error = 6.04%*This is the graph of the IBM stock that we took the information from to form our standard deviation to show how standard deviation isnt always the best way to represent the statistics of a particular set of data *Standard deviation is frequently used as a measure of the

20、volatility of a random financial variable*Here are some graphs of financial volatility *You can use the mean or median when trying to calculate the standard deviation if the data is normally distributed because they should both be very close*The Median is the middle number or term in a series of num

21、bers that are in order from least to greatest Ex. 2,2,2,3,3,3,4,4,5,6,6,7,7 The median is 4The mean would be Mean = Sum of data Total # pieces of data5413Mean= 4.15So the mean and median in this case are very similarQuestions1) Find the mean and standard deviation for this set of data.97, 100, 99, 9

22、5, 97, 101, 99#Data pieceData piece - MeanSquared Answer1234567Total=2. Find the mean for the given data:0.60, 0.94, 0.76, 1.02, 0.68, 0.72, 0.79, 0.92Mean =3. Mrs. Wells gave a math test to 12 random students the scores were as follows: 79, 84, 92, 78, 76, 85, 94, 89, 77, 81, 83, 96Find the mean an

23、d standard deviation for the test scores#Data PieceData piece - meanSquared answer123456789101112Total =4. A grade 3 test was given to 1000 randomly selected grade 10 students would you expect the results to be normally distributed, explain why or why not.5.ClassMean sleeping timesStandard deviation

24、18h1.2h210h2h39h0.9h47h1.5hA) A student slept for 12h, which class would you expect he is most likely from?B) Another student slept for 8h why is it hard to say which class he was form?6. A restaurant seats 162 people, average stay was 56 minutes, the standard deviation is 8 minutes. Would be unusua

25、l for some one to stay for one hour and 5 minute?7. These are the earnings of 9 families in a neighborhood: 27 000, 29 000, 25 000, 32 000, 30 500, 28 500, 35 000, 27 500, 31 000Find the standard deviation, mean, percentage error and label the bell curve#Data PieceData Piece - MeanSquared Answer1234

26、56789Total= 8. On average students get 9 hours of sleep per night, the standard deviation is 1.2 hours, fill in the blanksA) 68% of the students slept between _h and _hB) 95% of the students slept between _h and _h9. A sample of 10 silver maples height was conducted and they found the heights were a

27、s follows: 2.4m, 2m, 3.2m, 2.6m, 3.1m, 1.9m, 2.2m, 3.6m, 3.5m, and 2.1mFind the standard deviation, mean and percentage error#Data PieceData piece - MeanSquared Answer12345678910Total=10. If the average return from an investment of 1000 dollars is 275 dollars and the standard deviation is 15.5. How

28、likely is it to get a return of 315 dollars? How many standard deviations away from the mean would it be?11. A variance of 39.24 was found and there were 9 pieces of data find the standard deviation 12. Calculate the median of the following data:13,16,18,14,18,12,17,13,15,19,13,15,1413. On average s

29、tudents spend 4 hours on math homework per day. The standard deviation is 0.35A) What percent of students do math homework between 3.65 to 4.35 hours per night? B) What percent of math students do math homework between 3.3 to 4.7 hours per night 14. If you have a variance of 67 and the total number

30、of pieces of data are 25 what is your standard deviation?15. The average student tutorial kit was 20 pages with a standard deviation of 1.5A) 68% of the students tutorial kits were from _ to _ pages.B) 95% of the students tutorial kits were from _ to _ pages.Appendix#1Mean = Sum of data Total # piec

31、es of data6887Mean = 98.2#Data pieceData piece - MeanSquared Answer19797 - 98.2= -1.21.442100100 - 98.2= 1.83.2439999 - 98.2= 0.80.6449595 - 98.2= -3.210.2459797 98.2= -1.121.446101101 98.2= 2.87.8479999 98.2= 0.80.64Total= 25.48 Standard Deviation = (of squared total / # of pieces of data)Standard

32、Deviation = (3.64)Standard Deviation = 1.9#2Mean = Sum of data Total # pieces of data6.438Mean= 0.80#3Mean = Sum of data Total # pieces of data101412Mean= 84.5#Data PieceData piece - meanSquared answer17979 84.5= -5.530.2528484 84.5= -0.50.2539292 84.5= 7.556.2547878 84.5= -6.5 42.4557676 84.5= -8.5

33、72.2568585 84.5= 0.5.2579494 84.5= 9.590.2588989 84.5= 4.520.2597777 84.5= -7.556.25108181 84.5= -3.512.25118383 84.5= -1.52.25129696 84.5= 11.5132.25Total = 515.2Standard Deviation = (of squared total / # of pieces of data)Standard Deviation = (42.93)Standard Deviation = 6.55#4The data would not be

34、 evenly distributed because the data would be clustered around the 100% mark because it would be very easy to grade 10 students.#55a) The student would most likely be from class two because their average sleeping time is the highest5b) It is hard to tell because class one three and four and be in th

35、e first standard deviation away#6It is not very unusual#7Mean = Sum of data Total # pieces of data265 5009Mean= 29 500#Data PieceData Piece - MeanSquared Answer127 00027 000 - 29 500= -2 5006 250 000229 00029 000 - 29 500= -500250 000325 00025 000 - 29 500= -4 500 20 250 000432 00032 000 - 29 500= 2

36、 5006 250 000530 50030 500 - 29 500= 1 0001 000 000628 50028 500 - 29 500= -1 0001 000 000735 00035 000 - 29 500= 5 50030 250 000827 50027 500 - 29 500= -1 5002 250 000931 00031 000 - 29 500= 1 5002 250 000Total=69 750 000Standard Deviation = (of squared total / # of pieces of data)Standard Deviatio

37、n = (7 750 000)Standard Deviation = 2783.9Percentage error=(Standard Deviation / Mean) x 100Percentage error= (2783.9 / 29 500) x 100Percentage error= (.0943) x 100Percentage error= 9.43% 21148.3-23932.2-26716.1-29500-32283.9-35067.8-37851.7#8C) 68% of the students slept between 7.8h and 10.2hD) 95%

38、 of the students slept between _6.6h and _11.4_h#9Mean = Sum of data Total # pieces of data26.610Mean= 2.66#Data PieceData piece - MeanSquared Answer12.42.4 2.66= -0.260.067622.02.0 2.66= -0.660.43533.23.2 2.66= 0.540.29142.62.6 2.66= -0.060.003653.13.1 2.66= 0.440.19361.91.9 2.66= -0.760.55772.22.2

39、 2.66=-0.460.21183.63.6 2.66= -0.940.88393.53.5 2.66= 0.840.705102.12.1 2.66= -0.560.313Total= 3.61Standard Deviation = (of squared total / # of pieces of data)Standard Deviation = (0.361)Standard Deviation = 0.6Percentage error=(Standard Deviation / Mean) x 100Percentage error= (0.6 2.66) x 100Perc

40、entage error= (0.225) x 100Percentage error= 22.5%#10It is not very likely due to the fact that it is three standard deviations away#11Standard Deviation = (of squared total / # of pieces of data)Standard Deviation = (4.36)Standard Deviation = 2.08#12Unsorted data13,16,18,14,18,12,17,13,15,19,13,15,

41、14Sorted Data12, 13, 13, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19Median= 15#13A) 68%B) 95%#14Standard Deviation = (of squared total / # of pieces of data)Standard Deviation = (2.68)Standard Deviation = 1.64#15A) 68% of the students tutorial kits were from _18.5_ to _21.5_ pages.B) 95% of the students

42、tutorial kits were from 17_ to _23_ pages.Quiz1) 960 students took a quiz marked out of 175. The results were normally distributed and the mean was equal to 115 the standard deviation was 25; within what range would 68% of the students fall under?2) Someone claimed they got 170 on the same test. Wou

43、ld you believe them? Why or why not?3) The standard deviation is 2 and the mean is 6 label the given graph including the mean and each standard deviation4) What does this mean (å)?5) What is the difference between mean and median?6) The grades of all the boys in a math class was recorded and th

44、e results were as follows: 79, 88, 92, 86, 85, 82, 76, 87, 90, 83Find the mean and standard deviation #Data PieceData Piece - MeanSquared Answer12345678910Total= 7) What is the definition of variance?8) 10 runners each ran a 200-meter sprint, they all had different times they are: 30s, 29s, 34s, 33s, 31s, 28s, 30, 32s, 26s, 27sFind the Mean, standard deviation, percentage error and label the deviations on the graph provided#Data pieceData Piece - MeanSquared Answer12345678910Total=Quiz Answers#190 to 140#2It would be improbable but not impossible #3 0-2-4-6-8-10-12#4The sum of