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Math Exercises - Vietlist.US
Simple Statistics Review.

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Mean   = ∑ data points =    Total
number of data points        n       

Median:
Test results: 1, 2, 5, 9, 11 --> Median = 5
Test results: 3, 4, 5, 6, 8, 10 --> Median = (5+6)/2 = 5.5

Range:
Test results: 1, 2, 3, 5, 8, 10 --> Range = 10 - 1 = 9

Error  =  Experimental value - 'true' or theoretical value

Percent Error   = Error x  100
Theoretical value

Deviation  =   Experimental value - Mean

Percent Deviation   = Deviation x  100
  Theoretical value  

Standard Deviation

Standard deviation (σ) is a particularly useful tool in helping to check the spreading of the data. The bell curve is the most common type of the distribution for a variable. One standard deviation, or one sigma, away from the mean in either direction on the horizontal axis accounts for around 68 percent of the data points. Two standard deviations, or two sigmas, away from the mean account for roughly 95 percent of the data points. Three standard deviations account for about 99 percent of the data points.

σ   =   ∑ d2  
  n - 1  

bell curve
The Bell Curve

Calculation using the z-score method:
First calculate the z-score and then look up its corresponding p-value using the standard normal table.

The Standard Normal Table

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0 0.5 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.5279 0.53188 0.53586
0.1 0.53983 0.5438 0.54776 0.55172 0.55567 0.55962 0.56356 0.56749 0.57142 0.57535
0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409
0.3 0.61791 0.62172 0.62552 0.6293 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173
0.4 0.65542 0.6591 0.66276 0.6664 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793
0.5 0.69146 0.69497 0.69847 0.70194 0.7054 0.70884 0.71226 0.71566 0.71904 0.7224
0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.7549
0.7 0.75804 0.76115 0.76424 0.7673 0.77035 0.77337 0.77637 0.77935 0.7823 0.78524
0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327
0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891
1 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214
1.1 0.86433 0.8665 0.86864 0.87076 0.87286 0.87493 0.87698 0.879 0.881 0.88298
1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147
1.3 0.9032 0.9049 0.90658 0.90824 0.90988 0.91149 0.91308 0.91466 0.91621 0.91774
1.4 0.91924 0.92073 0.9222 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189
1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408
1.6 0.9452 0.9463 0.94738 0.94845 0.9495 0.95053 0.95154 0.95254 0.95352 0.95449
1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.9608 0.96164 0.96246 0.96327
1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062
1.9 0.97128 0.97193 0.97257 0.9732 0.97381 0.97441 0.975 0.97558 0.97615 0.9767
2 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.9803 0.98077 0.98124 0.98169
2.1 0.98214 0.98257 0.983 0.98341 0.98382 0.98422 0.98461 0.985 0.98537 0.98574
2.2 0.9861 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.9884 0.9887 0.98899
2.3 0.98928 0.98956 0.98983 0.9901 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158
2.4 0.9918 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361
2.5 0.99379 0.99396 0.99413 0.9943 0.99446 0.99461 0.99477 0.99492 0.99506 0.9952
2.6 0.99534 0.99547 0.9956 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643
2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.9972 0.99728 0.99736
2.8 0.99744 0.99752 0.9976 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807
2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861
3 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.999
3.1 0.99903 0.99906 0.9991 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929
3.2 0.99931 0.99934 0.99936 0.99938 0.9994 0.99942 0.99944 0.99946 0.99948 0.9995
3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.9996 0.99961 0.99962 0.99964 0.99965
3.4 0.99966 0.99968 0.99969 0.9997 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976
3.5 0.99977 0.99978 0.99978 0.99979 0.9998 0.99981 0.99981 0.99982 0.99983 0.99983
3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989
3.7 0.99989 0.9999 0.9999 0.9999 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992
3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995
3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997
4 0.99997 0.99997 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998

Example of calculation using the z-score method:

If the z-score = 1.25. Follow the rows down to 1.2 and then across the columns to 0.05. The P-value is the highlighted box with a value of 0.89435. Values in the table represent area under the standard normal distribution curve to the left of the z-score.

Using the previous example:

Z-score = 1.25, P-value = 0.89435 = 89.435% is graphically represented below.

StandardTable

bell-curve

The area of 89.435% is corresponding to z = 1.25. Remember that the area starts from 0 to the point of 1.25 sigma.

 

 

 

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