STATISTICSIMPORTANT FORMULAE AND POINTS TO REMEMBERSTATISTICS is the Art and Science of gathering, analyzing and making inferences (predictions)from numerical information obtained in an experiment. This numerical information is referredto as Data.An AVERAGE is a number that is a representative of a group of data. There are atleast fourDifferent averages the mean, the median, the mode and the midrange.AVERAGES are commonly referred to as Measures of Central Tendency. 1. The Average or the arithmetic mean: If x1, x2, x3, ….xn are n given observations, then Arithmetic mean _ (X ) = x1+x2+x3….xn (for ungrouped data)n 2. For grouped data: (DIRECT METHOD) _ Mean X = ∑fx(where f is the frequency) ∑f 3. For grouped data: (SHORT-CUT METHOD) _Mean X= A+ ∑fd where A is the Assumed mean & d= X-A∑f 4. Step-deviation Method: _ Mean X = A + h X ∑fu∑fWhere A = Assumed mean, D=X–Ah = Choose a number ‘h’ which is common factor of all ‘d’ s. U= X–A H
5. MEDIAN: It is the value which divides the data into two equal parts- the middle most observation. The data is arranged in ascending or descending order and the middle item is the median. The median of raw data (ungrouped) when the number of observations n is odd, is the(n + 1)th observation. 2 When the number of observations is even, then it is the average of the nth and (n+1)th 2 2Observation. 6. Median for continuous frequency distribution: (grouped data) Median = L + (N- M) C 2fwhere L is the lower limit of the median classC be the width of the class intervalM be the cumulative frequency of the class just before the median class. F be the frequency of the median class. 7. MODE: Mode is the value of the variable in the data which occurs most frequently i.e, the mode isThat observation which has maximum frequency. MODE = L + (fm – fl) x h2 fm- f1-f2Where L = lower limit of modal class ( class with maximum frequency) Fm= Frequency of the class preceding the modal class. f1 = Frequency of the class preceding the modal class.
f2 = Frequancy of the class succeeding the modal class.h = width of the modal class. 8. RANGE: Range is the difference between the greatest and the least of the observations. But there are other and better measures of dispersion called as STANDARD DEVIATION. Standard deviation value is based upon each and every item of the data. It is denoted by The greek letter (sigma) 9. STANDARD DEVIATION FOR AN UNGROUPED DATA:1. Standard deviation _ _Where d = X - X &X is the mean 2. Standard deviation Where d = X – A where A is the assumed mean. 10. STANDARD DEVIATION FOR DISCRETE SERIES: 1. When deviations are taken from Arithmetic mean.Where d = X –X and X is the mean 2. When deviations are taken from assumed meanWhere d= X-A, A being assumed mean. 11. VARIANCE: The mean of the squares of the deviations of the values of the variable is called Variance
Variance and it is denoted by ∑2 Variance is the square of the standard deviation, The positive square root of the Variance is defined as standard deviation. S.D. of the first ‘n’ natural number isNOTE: The standard deviation of a series remains unchanged when each value is Added or subtracted by the same quantity. The standard deviation of a series gets multiplied or divided by the Quantity K if each value is multipled (or) divided by K. PROBABILITY The word probability and chance are quite familiar to every one. In order to think about and measure uncertainity, we turn to a branch of Mathematics called “ PROBABILITY”. RANDOM EXPERIMENT: Any experiment whose outcomes cannot be predicted in advance or Determine in advance is a rancom experiment. TRIAL: Each performance of the random experiment is called a trial. SAMPLE SPACE: The set of all possible outcomes of a random experiment is called a sample Space denoted by S. SAMPLE POINT: Each element of the sample space is called a sample point. EVENT: An event is a subset of a sample space.
EQUALLY LIKELY EVENTS: Two or more events are said to be equally likely if each one of them has an Equal chance of occurring. (In tossing a coin, getting a head and getting a tail are equally likely events) MUTUALLY EXCLUSIVE EVENTS: Two events A & B are said to be mutually exclusive events if the occurrence of any one Of them excludes the occurrence of the other even. i.e, they cannot occur Simultaneously. FAVOURABLE EVENTS OR CASES: The number of outcomes or cases which entail the occurrence of the event In an experiment are called favourable events or favourable cases. PROBABILITY: If A is the favourable event, which is a subset of the sample space S of the Experiment, then the probability of A denoted by P(A) is defined by, P(A) = n(A) n(S) Where n (A), n (S) are the cardinal number of Sets A and S provided the Members of the Set S are equally likely. NOTE: 1. O ≤ P (A) ≤ 1 2. P (A) = 1- P (A) Therefore P(A) + P(A) = 1 3. If P(A) = 0, then A is an impossible event. i.e, Probability of an impossible Event is 0i.e, p (Ø) = 0 4. Probability of the sure event is 1. i.e, P (S) = 1 , S is called Sure event.ADDITION THEOREM ON PROBABILITY: 1. If A and B are any two events, then P (AUB) = P (A) + P (B) - P (AΩB) 2. If A and B are mutually exclusive events, then P (AÙB) = P (A) + P (B)================================================================