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Statistics Formulae for School Students

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FIND THE FORMULA STATISTICS FORMULAE FOR SCHOOL STUDENTS M. D. Raghu Ganit learning guides
Contents 1. Data spread, Central values. 2. Box and Whiskers diagram 3. Histogram 4. Probability rules 5 Tree diagram 6 Poisson probability 7. Binomial probability 8. Normal distribution curves 9. Standard normal distribution 10. Normal distribution table 11 Correlation 12 Regression line 13 Central limit theorem 14. Confidence interval and level 15. Confidence interval table 16. Expected value
Data spread, Central values Individual data spread Mean 亮 = x/n Variance v = [(x亮)族]/n Standard deviation v = = s Mode = Most common value. Median = Middle value n = Population size Group data spread Mean 亮 = (f)/裡f Population variance . v= f( 亮)族/f. Standard deviation s = v f = Class frequency = Class mid-range 裡f = Total frequency
Box and whisker diagram Box and whisker chart Chart properties A box plot represents data divided into four quartiles. Difference between maximum and minimum values are divided into two whiskers and a box . Each whisker represents upper or lower quartile and the box inter quartile. A line within the box shows the median.
Histogram Histogram chart Data representation A histogram is a graphical representation of data by columns. Column width represents a class interval. Height of a column is proportional to the data frequency As an alternative, area of each box can be made to represent the frequency.
Probability rules P(A) = Probability of an event A P(A) is favourable outcomes to total outcomes ratio P(A) = nf /nt. P(A) is unfavorable outcomes to total outcomes ratio. P(A) = 1-P(A) If A and B are independent events then probability of A or B is called Probability of A union B. P(AB)=P(A)+P(B) If A and B are not independent P(AB)=P(A)+P(B)-P(AB) P(AB) is the occurrence of events A and B together hence, P(AB)=P(A)P(B given A).
Tree diagram Tree diagram is a simple representation of probabilities. The branches show outcomes of the first event followed by outcomes of further events. Tree diagrams can be used for both dependent and independent events. Serial branches gives probability combinations and diverging branches gives probability sums.
Poisson probability distribution Distribution probability Probability conditions Poisson distribution Is a discreet probability of a number of events over a specified interval. Interval is an unit such as time, distance or number of people. 了 is a relatively small number as compared to the population considered.
Binomial probability distribution Probability function Binomial probability Binomial probability is for a distribution that has only two types of outcomes. The probability that in n number of trials there are r desired outcomes is given by P(R). Probabilities of desired and undesired out comes add up to 1. Binomial probability is a discrete distribution.
Normal distribution curves Bell shaped curve Normal distribution properties Normal distribution is a bell shaped curve. This curve follows Gaussian distribution pattern. Measured statistical parameter, for example height of students in a school, is shown on the X axis. Population or Probability density is shown on Y axis. Area under the entire curve covers the total population or a Probability of 1. Part of the curve gives the part of population in between any two x values Ratio of part of population between two x values to the total population gives the Probability of a parameter x given x<x<x to be within x and x . Normal distribution is a method for continuous data Probability analysis.
Standard normal distribution Bell shaped curve Normal probability distribution X axis of a normal distribution curve gives the population away from the mean. Area under any part of the curve is a measure of the population probability. Standardised X axis represents Z score. Percentage area under the curve for any two Z scores gives the probability. Z = (x 亮)/, =standard deviation, 亮=mean Y=Probability density=[e^(-z族/2)]/(2)
Standard normal distribution table Normal distribution curve Z score vs. probability table P(Z=Zn<Zn+1) % Probability P(Z=-3<-2) 2.1 P (Z=-2<-1) 13.6 P(Z=-1<0) 34.1 P(Z=0<1) 34.1 P(Z=1<2) 13.6 P(Z=2<3) 2.1
Correlation Regression factor r X and Y correlation Correlation is a statistical relationship between two sets of variables x and y. Correlation or regression factor r is calculated from the sets of variables x,y and their means 亮x, 亮y. r = 賊1 indicates a strong correlation between x and y and no correlation if r = 0.
Regression line Regression Line equation Regression line is the best fit line for a scatter plot. r is the correlation factor. Line equation: y = mx + c where m = r(s甬/s) and c= 亮甬 - m(亮) s is the standard deviation and 亮 is the mean. Regression Line graph
Central limit theorem Central limit theorem is about samples of size 単 taken from a large population n with mean 亮 and standard deviation s. The theorem is valid for normal or any other type of distribution. This theorem states that the distribution of the sample means x will be close to a normal distribution. x is the mean of a sample set. Mean 亮x of the sample means will be equal to the mean 亮 of the population. Standard deviation s of the x values will be equal to s/単 The random samples are taken from the population, with replacements. It should be sufficiently large. At least 30 or more samples are required.
Confidence interval and level Confidence interval is a range of values that is likely to contain a population parameter such as the mean. Confidence interval is expressed as a range. If a parameter is obtained from a sample survey then the same parameter for the population will probably lie with in this range. Confidence level is the percentage probability of a population parameter to lie with in a range of variation called confidence interval. For example, consider average scores of a sample group of students. One can say with 90% confidence level that the average score of entire school students is likely to be with in 50 to 80 marks confidence interval.
Confidence interval table Confidence interval Ci x+Zs/n < 亮 < xZs/n 亮 = population mean x = sample means. Z = z score for desired level (CL) of 亮 as per CL versus Z table. s = large sample or population standard deviation n = sample size Margin of error = 賊Z(s/n) Confidence percentage table CL Z. 90%. 1.645 95%. 1.960 98%. 2.326 99%. 2.576. 99.9%. 3.291 CL = Confidence level.
Expected value Value function E.V. = E(X) = 裡xP(X=x) = 亮 = E(X族) [E(X)]族. 亮=mean, =standard deviation If c and k are constants, E(c+kX) = c + k(EX) Expected value of x and w E(X+W) = E(X)+ E(W) Definition Expected value EV is the sum of each possible outcome multiplied by the probability of each outcome. Expected value EV is also the mean value 亮 of the outcomes. Expected values of combination of two variables Is the sum of expected value of each variable.
GANIT LEARNING GUIDES Thank you M. D. Raghu For mathematics and physics formulae 際際滷Share.net/dhatiraghu

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Statistics Formulae for School Students

  • 1. FIND THE FORMULA STATISTICS FORMULAE FOR SCHOOL STUDENTS M. D. Raghu Ganit learning guides
  • 2. Contents 1. Data spread, Central values. 2. Box and Whiskers diagram 3. Histogram 4. Probability rules 5 Tree diagram 6 Poisson probability 7. Binomial probability 8. Normal distribution curves 9. Standard normal distribution 10. Normal distribution table 11 Correlation 12 Regression line 13 Central limit theorem 14. Confidence interval and level 15. Confidence interval table 16. Expected value
  • 3. Data spread, Central values Individual data spread Mean 亮 = x/n Variance v = [(x亮)族]/n Standard deviation v = = s Mode = Most common value. Median = Middle value n = Population size Group data spread Mean 亮 = (f)/裡f Population variance . v= f( 亮)族/f. Standard deviation s = v f = Class frequency = Class mid-range 裡f = Total frequency
  • 4. Box and whisker diagram Box and whisker chart Chart properties A box plot represents data divided into four quartiles. Difference between maximum and minimum values are divided into two whiskers and a box . Each whisker represents upper or lower quartile and the box inter quartile. A line within the box shows the median.
  • 5. Histogram Histogram chart Data representation A histogram is a graphical representation of data by columns. Column width represents a class interval. Height of a column is proportional to the data frequency As an alternative, area of each box can be made to represent the frequency.
  • 6. Probability rules P(A) = Probability of an event A P(A) is favourable outcomes to total outcomes ratio P(A) = nf /nt. P(A) is unfavorable outcomes to total outcomes ratio. P(A) = 1-P(A) If A and B are independent events then probability of A or B is called Probability of A union B. P(AB)=P(A)+P(B) If A and B are not independent P(AB)=P(A)+P(B)-P(AB) P(AB) is the occurrence of events A and B together hence, P(AB)=P(A)P(B given A).
  • 7. Tree diagram Tree diagram is a simple representation of probabilities. The branches show outcomes of the first event followed by outcomes of further events. Tree diagrams can be used for both dependent and independent events. Serial branches gives probability combinations and diverging branches gives probability sums.
  • 8. Poisson probability distribution Distribution probability Probability conditions Poisson distribution Is a discreet probability of a number of events over a specified interval. Interval is an unit such as time, distance or number of people. 了 is a relatively small number as compared to the population considered.
  • 9. Binomial probability distribution Probability function Binomial probability Binomial probability is for a distribution that has only two types of outcomes. The probability that in n number of trials there are r desired outcomes is given by P(R). Probabilities of desired and undesired out comes add up to 1. Binomial probability is a discrete distribution.
  • 10. Normal distribution curves Bell shaped curve Normal distribution properties Normal distribution is a bell shaped curve. This curve follows Gaussian distribution pattern. Measured statistical parameter, for example height of students in a school, is shown on the X axis. Population or Probability density is shown on Y axis. Area under the entire curve covers the total population or a Probability of 1. Part of the curve gives the part of population in between any two x values Ratio of part of population between two x values to the total population gives the Probability of a parameter x given x<x<x to be within x and x . Normal distribution is a method for continuous data Probability analysis.
  • 11. Standard normal distribution Bell shaped curve Normal probability distribution X axis of a normal distribution curve gives the population away from the mean. Area under any part of the curve is a measure of the population probability. Standardised X axis represents Z score. Percentage area under the curve for any two Z scores gives the probability. Z = (x 亮)/, =standard deviation, 亮=mean Y=Probability density=[e^(-z族/2)]/(2)
  • 12. Standard normal distribution table Normal distribution curve Z score vs. probability table P(Z=Zn<Zn+1) % Probability P(Z=-3<-2) 2.1 P (Z=-2<-1) 13.6 P(Z=-1<0) 34.1 P(Z=0<1) 34.1 P(Z=1<2) 13.6 P(Z=2<3) 2.1
  • 13. Correlation Regression factor r X and Y correlation Correlation is a statistical relationship between two sets of variables x and y. Correlation or regression factor r is calculated from the sets of variables x,y and their means 亮x, 亮y. r = 賊1 indicates a strong correlation between x and y and no correlation if r = 0.
  • 14. Regression line Regression Line equation Regression line is the best fit line for a scatter plot. r is the correlation factor. Line equation: y = mx + c where m = r(s甬/s) and c= 亮甬 - m(亮) s is the standard deviation and 亮 is the mean. Regression Line graph
  • 15. Central limit theorem Central limit theorem is about samples of size 単 taken from a large population n with mean 亮 and standard deviation s. The theorem is valid for normal or any other type of distribution. This theorem states that the distribution of the sample means x will be close to a normal distribution. x is the mean of a sample set. Mean 亮x of the sample means will be equal to the mean 亮 of the population. Standard deviation s of the x values will be equal to s/単 The random samples are taken from the population, with replacements. It should be sufficiently large. At least 30 or more samples are required.
  • 16. Confidence interval and level Confidence interval is a range of values that is likely to contain a population parameter such as the mean. Confidence interval is expressed as a range. If a parameter is obtained from a sample survey then the same parameter for the population will probably lie with in this range. Confidence level is the percentage probability of a population parameter to lie with in a range of variation called confidence interval. For example, consider average scores of a sample group of students. One can say with 90% confidence level that the average score of entire school students is likely to be with in 50 to 80 marks confidence interval.
  • 17. Confidence interval table Confidence interval Ci x+Zs/n < 亮 < xZs/n 亮 = population mean x = sample means. Z = z score for desired level (CL) of 亮 as per CL versus Z table. s = large sample or population standard deviation n = sample size Margin of error = 賊Z(s/n) Confidence percentage table CL Z. 90%. 1.645 95%. 1.960 98%. 2.326 99%. 2.576. 99.9%. 3.291 CL = Confidence level.
  • 18. Expected value Value function E.V. = E(X) = 裡xP(X=x) = 亮 = E(X族) [E(X)]族. 亮=mean, =standard deviation If c and k are constants, E(c+kX) = c + k(EX) Expected value of x and w E(X+W) = E(X)+ E(W) Definition Expected value EV is the sum of each possible outcome multiplied by the probability of each outcome. Expected value EV is also the mean value 亮 of the outcomes. Expected values of combination of two variables Is the sum of expected value of each variable.
  • 19. GANIT LEARNING GUIDES Thank you M. D. Raghu For mathematics and physics formulae 際際滷Share.net/dhatiraghu