STATISTICSSTATISTICS

STATISTICSSTATISTICS

The numerical records of any event or The numerical records of any event or phenomena are referred to as phenomena are referred to as statistics. The data are the details in statistics. The data are the details in the numerical records or reports.the numerical records or reports.

TYPES OF STATISTICSTYPES OF STATISTICS

DESCRIPTIVE – It is concerned with the DESCRIPTIVE – It is concerned with the group of individuals actually observed. group of individuals actually observed. Statistics computed from these data or Statistics computed from these data or observations are called descriptive observations are called descriptive statistics.statistics.

INFERENTIAL- This is used to search for INFERENTIAL- This is used to search for general principles to be applied to a much general principles to be applied to a much larger domain beyond the group actually larger domain beyond the group actually observed. observed.

NEED OF STATISTICSNEED OF STATISTICS

HELPFUL TO A TEACHER- It helps teachers HELPFUL TO A TEACHER- It helps teachers in classifying a particular class into sub-in classifying a particular class into sub-groups for deciding about the method of groups for deciding about the method of teaching. Statistical methods are applied teaching. Statistical methods are applied to get information from the available data to get information from the available data to solve the practical problems.to solve the practical problems.

TOOL FOR SCIENTISTS- Psychologist take TOOL FOR SCIENTISTS- Psychologist take help of statistics for interpretation of help of statistics for interpretation of behavioural data . In research work behavioural data . In research work whether we explore new relationships or whether we explore new relationships or test theories our methods are largely test theories our methods are largely statistical.statistical.

DATADATA

The data are the details in the The data are the details in the numerical reports or records. The numerical reports or records. The word data is a plural term as it word data is a plural term as it usually refers to more than one usually refers to more than one observation or measure. observation or measure.

TYPES OF DATATYPES OF DATA CONTINUOUS – We can have any number of sub-CONTINUOUS – We can have any number of sub-

divisions in a continuous series and it is possible to divisions in a continuous series and it is possible to get infinite number of in-between values very near get infinite number of in-between values very near to each other in the series. The physical measures to each other in the series. The physical measures such as metres, litres,Kilograms,hours and mental such as metres, litres,Kilograms,hours and mental measures such as IQ fall into continuous series.measures such as IQ fall into continuous series.

DISCRETE – In discrete data measurements are DISCRETE – In discrete data measurements are expressed in whole units e.g. number of members expressed in whole units e.g. number of members in a family, married and unmarried in a family, married and unmarried persons,number of girl and boy students in a class.persons,number of girl and boy students in a class.

SCORESCORE

A score is described as an interval which A score is described as an interval which spreads from 5 unit below to 5 unit above the spreads from 5 unit below to 5 unit above the score. In a mental test, a score is a unit score. In a mental test, a score is a unit distance between the lower limit and the distance between the lower limit and the upper limit e.g. a score of 110 in an upper limit e.g. a score of 110 in an intelligence test denotes the score interval intelligence test denotes the score interval 109.5 to 110.5, the exact midpoint of which is 109.5 to 110.5, the exact midpoint of which is the score itself 110.the score itself 110.

109.5109.5 110110 110.5110.5

FREQUENCY DISTRIBUTIONFREQUENCY DISTRIBUTION

Organizing the data by grouping the scores Organizing the data by grouping the scores into categories of scores. When the number into categories of scores. When the number of measurements is small we keep the data of measurements is small we keep the data ungrouped. When it is large we need to ungrouped. When it is large we need to group it into a frequency distribution.group it into a frequency distribution.

By preparing Frequency distribution we By preparing Frequency distribution we arrange the data in systematic order by arrange the data in systematic order by grouping the scores into classes.grouping the scores into classes.

The data become more organized, The data become more organized, meaningful and Interpretable.meaningful and Interpretable.

STEPS IN PREPARING THE FREQUENCY STEPS IN PREPARING THE FREQUENCY DISTRIBUTIONDISTRIBUTION

DETERMINE THE RANGE – The range is defined as the DETERMINE THE RANGE – The range is defined as the highest score minus the lowest score. e.g. highest score is highest score minus the lowest score. e.g. highest score is 86 and the lowest score is 22 the range is 64.86 and the lowest score is 22 the range is 64.

DECIDE THE SIZE OF INTERVAL – The second step is to DECIDE THE SIZE OF INTERVAL – The second step is to decide the size of the class interval or no. of groupings to decide the size of the class interval or no. of groupings to be used in the distribution. The most commonly used size be used in the distribution. The most commonly used size of class interval is 3,5,10 units.of class interval is 3,5,10 units.

NUMBER OF CLASS INTERVALS – The number of class NUMBER OF CLASS INTERVALS – The number of class intervals can be known by dividing the range by the size of intervals can be known by dividing the range by the size of the interval e.g. the range is 64 when divided by 5 it gives the interval e.g. the range is 64 when divided by 5 it gives 12.8 or 13 class intervals.12.8 or 13 class intervals.

TALLYING THE SCORES – The practice is to take one score TALLYING THE SCORES – The practice is to take one score at a time and record a tally mark for it against its at a time and record a tally mark for it against its appropriate interval. appropriate interval.

SUMMING UP TALLIES AND FINDING OUT N – The sum of all SUMMING UP TALLIES AND FINDING OUT N – The sum of all the numbers in the frequency colomn gives N. the numbers in the frequency colomn gives N.

EXAMPLEEXAMPLECLASS INTERVAL ( CI)CLASS INTERVAL ( CI) TALLY MARKSTALLY MARKS FREQUENCY (f)FREQUENCY (f)85-8985-89 //// 2280-8480-84 ////// 3375-7975-79 //////// 5570-7470-74 //// 2265-6965-69 ////// 3360-6460-64 //// //// 5555-5955-59 //// /////// /// 8850-5450-54 //////// 4445-4945-49 //////// 5540-4440-44 //////// 4435-3935-39 ////// 3330-3430-34 //////// 444425-2925-29 00 0020-2420-24 //// 22

5050

EXERCISEEXERCISE

Q1. For the following data draw Q1. For the following data draw frequency distribution using a class frequency distribution using a class interval of 3 and 5.interval of 3 and 5.

Marks of psychology students of class Marks of psychology students of class XIXI

72,69,84,82,67,73,72,63,78,81,70,76,772,69,84,82,67,73,72,63,78,81,70,76,76,75,72,72,86,65,71,83,64,67,77,67,6,75,72,72,86,65,71,83,64,67,77,67,61.61.

MEASURES OF CENTRAL MEASURES OF CENTRAL TENDENCYTENDENCY

The measures of central tendency represent The measures of central tendency represent the central values of a group of individuals.the central values of a group of individuals.

The important way of describing the scores of The important way of describing the scores of individuals in a group are by averages.individuals in a group are by averages.

The average stands for any measure of The average stands for any measure of central tendency.central tendency.

The commonly used measures of central The commonly used measures of central tendency are Mean, Median and Mode.tendency are Mean, Median and Mode.

USES OF CENTRAL TENDENCY MEASURESUSES OF CENTRAL TENDENCY MEASURES

Indicator of group performanceIndicator of group performance – A mass of individual scores – A mass of individual scores of a group is reduced to a single score description by a of a group is reduced to a single score description by a measure of central tendency or an average. An average measure of central tendency or an average. An average represents all the scores obtained by the group of represents all the scores obtained by the group of individuals it is economical and meaningful. It is descriptive individuals it is economical and meaningful. It is descriptive of the performance of group as a whole.of the performance of group as a whole.

Comparability of two or more groups Comparability of two or more groups – By the use of – By the use of averages we are able to compare two or more groups with averages we are able to compare two or more groups with regard to their typical performance.regard to their typical performance.

Estimates of population averagesEstimates of population averages – Sample averages are – Sample averages are close estimates of population averages. With our sample close estimates of population averages. With our sample average we can generalize beyond samples and make average we can generalize beyond samples and make predictions for the population. This is possible when our predictions for the population. This is possible when our sample properly represents its population.sample properly represents its population.

THE MEAN OR ARTHMETIC THE MEAN OR ARTHMETIC MEANMEAN

Mean is the arithmetic average of the Mean is the arithmetic average of the individual scores, it is the sum of the individual scores, it is the sum of the separate scores divided by their number.separate scores divided by their number.

The Mean is denoted by MThe Mean is denoted by M(Here we will calculate Mean by three (Here we will calculate Mean by three methods)methods)

Mean of Un grouped dataMean of Un grouped data Mean of grouped data by long – methodMean of grouped data by long – method Mean of grouped data by shortcut method Mean of grouped data by shortcut method

or assumed mean methodor assumed mean method

EXAMPLEEXAMPLE Suppose five students of the group have scored following marks in Suppose five students of the group have scored following marks in

psychology.psychology.X1= 53, X2 = 67, X3= 65, X4 = 58, X5= 62.X1= 53, X2 = 67, X3= 65, X4 = 58, X5= 62. M =M = € X € X

NNWhere, M= MeanWhere, M= Mean € € = Sum total of= Sum total of X = each individual scoreX = each individual score N = Number of measuresN = Number of measuresM = M = X1+X2+X3+X4+X5X1+X2+X3+X4+X5 NN = = 53+67+65+58+6253+67+65+58+62 55 == 305 305 55 = 61= 61

Mean of ungrouped dataMean of ungrouped data

Formulae of Mean for ungrouped data.Formulae of Mean for ungrouped data. M =M = € X € X

NN

Where, M= MeanWhere, M= Mean

€ € = Sum total of= Sum total of

X = each individual scoreX = each individual score

N = Number of measuresN = Number of measures

EXERCISEEXERCISE

Q1. For the following data calculate MeanQ1. For the following data calculate MeanMarks of psychology students of class XIMarks of psychology students of class XI72,69,84,82,67,73,72,63,78,81,70,76,76,75,72,69,84,82,67,73,72,63,78,81,70,76,76,75,

72,72,86,65,71,83,64,67,77,67,61.72,72,86,65,71,83,64,67,77,67,61.Q2.Calculate Mean for the marks obtained Q2.Calculate Mean for the marks obtained

by 10 students in Englishby 10 students in English78, 24, 49,99,72,88,56,66,42,6278, 24, 49,99,72,88,56,66,42,62Q3. Calculate Mean of Marks obtained by 8 Q3. Calculate Mean of Marks obtained by 8

students in Physics.students in Physics.72,81,69,78,74,91,88,89 72,81,69,78,74,91,88,89

MEAN OF GROUPED DATA BY LONG METHODMEAN OF GROUPED DATA BY LONG METHOD When the scores are too many we group them When the scores are too many we group them

into a frequency distribution and apply any into a frequency distribution and apply any formula to compute the mean.formula to compute the mean.

M =M = €fX €fXNN

Where,Where,M = meanM = mean f = frequency of class intervalsf = frequency of class intervals X= Midpoints of class intervalX= Midpoints of class interval N = Number of measuresN = Number of measures

ExampleExample

Class interval (C.I mid-points f fXClass interval (C.I mid-points f fX85-8985-89 87 87 22 17417480-8480-84 82 82 33 24624675-7975-79 77 77 55 38538570-7470-74 72 72 22 14414465-6965-69 67 67 33 20120160-6460-64 62 62 55 31031055-5955-59 57 57 88 456456

50-5450-54 52 52 44 20820845-4945-49 47 47 55 23523540-4440-44 42 42 44 16816835-3935-39 37 37 33 11111130-3430-34 32 32 44 12812825-2925-29 27 27 00 0 020-2420-24 22 22 22 4444

EXERCISEEXERCISE

Q1. Calculate Mean of the following data using long method.Q1. Calculate Mean of the following data using long method.SCORESSCORES FREQUENCYFREQUENCY60-6460-64 0055-5955-59 2250-5450-54 252545-4945-49 4848 40-4440-44 474735-3935-39 191930-3430-34 262625-2925-29 151520-2420-24 9915-1915-19 7710-1410-14 225-95-9 000-40-4 00

N = 200N = 200

EXERCISEEXERCISE

Q2. Calculate Mean of the following data using long method.Q2. Calculate Mean of the following data using long method.SCORESSCORES FREQUENCYFREQUENCY75-7975-79 1170-7470-74 3365-6965-69 6660-6460-64 1212 55-5955-59 202050-5450-54 363645-4945-49 202040-4440-44 151535-3935-39 6630-3430-34 4425-2925-29 22

N = 125N = 125

EXERCISEEXERCISE

Q2. Calculate Mean of the following data using long method.Q2. Calculate Mean of the following data using long method.SCORESSCORES FREQUENCYFREQUENCY195-199195-199 11190-194190-194 22185-189185-189 44180-184180-184 55 175-179175-179 88170-174170-174 1010165-169165-169 66160-164160-164 44155-159155-159 44150-154150-154 22145-149145-149 33140-144140-144 11

N = 50N = 50

MEAN OF GROUPED DATA BY SHORT-CUT MEAN OF GROUPED DATA BY SHORT-CUT METHOD OR ASSUMED MEAN METHODMETHOD OR ASSUMED MEAN METHOD

When the scores are too many we group them into a frequency distribution When the scores are too many we group them into a frequency distribution and apply formula of long method to compute the mean but sometimes data and apply formula of long method to compute the mean but sometimes data is too large to apply long method then we use assumed mean method.is too large to apply long method then we use assumed mean method.

M = AM + ciM = AM + ci

Where,Where,AM = Assumed meanAM = Assumed mean c = c = €fx’€fx’ NNWhere,Where,f = frequency of class intervalsf = frequency of class intervals x’= Deviation of scores from the assumed mean.x’= Deviation of scores from the assumed mean. N = Number of measuresN = Number of measures

i = Size of Class- Intervali = Size of Class- Interval

ExampleExample

Class intervalClass interval f f x’x’ fx’fx’

85-8985-89 2 2 66 121280-8480-84 3 3 55 151575-7975-79 5 5 44 202070-7470-74 2 2 33 66 646465-6965-69 3 3 22 6660-6460-64 5 5 11 5555-5955-59 8 8 00 00

50-5450-54 4 4 -1-1 -4-445-4945-49 5 5 -2-2 -10-1040-4440-44 4 4 -3-3 -12-1235-3935-39 3 3 -4-4 -12-12 -72-7230-3430-34 4 4 -5-5 -20-2025-2925-29 0 0 -6-6 0 020-2420-24 2 2 -7-7 -14-14

N = 50N = 50

EXERCISEEXERCISE

Q2. Calculate Mean of the following data using short-cut method or assumed mean Q2. Calculate Mean of the following data using short-cut method or assumed mean methodmethod

SCORESSCORES FREQUENCYFREQUENCY195-199195-199 11190-194190-194 22185-189185-189 44180-184180-184 55 175-179175-179 88170-174170-174 1010165-169165-169 66160-164160-164 44155-159155-159 44150-154150-154 22145-149145-149 33140-144140-144 11

N = 50N = 50

EXERCISEEXERCISE

Q2. Calculate Mean of the following data using short-cut method or assumed mean Q2. Calculate Mean of the following data using short-cut method or assumed mean method.method.

SCORESSCORES FREQUENCYFREQUENCY75-7975-79 1170-7470-74 3365-6965-69 6660-6460-64 1212 55-5955-59 202050-5450-54 363645-4945-49 202040-4440-44 151535-3935-39 6630-3430-34 4425-2925-29 22

N = 125N = 125

EXERCISEEXERCISE

Q1. Calculate Mean of the following data using short-cut method or assumed mean Q1. Calculate Mean of the following data using short-cut method or assumed mean methodmethodSCORESSCORES FREQUENCYFREQUENCY60-6460-64 0055-5955-59 2250-5450-54 252545-4945-49 4848 40-4440-44 474735-3935-39 191930-3430-34 262625-2925-29 151520-2420-24 9915-1915-19 7710-1410-14 225-95-9 000-40-4 00

N = 200N = 200

USE OF THE MEANUSE OF THE MEAN

When the central point having When the central point having maximum stability is needed.maximum stability is needed.

When the scores symmetrically fall When the scores symmetrically fall around a central point i.e. the around a central point i.e. the distribution of scores is not skewed.distribution of scores is not skewed.

When further calculations like When further calculations like standard deviation and correlation standard deviation and correlation coefficient are to be done.coefficient are to be done.

USE OF THE MEDIANUSE OF THE MEDIAN

When the exact 50 per cent point or the When the exact 50 per cent point or the mid-point of the distribution is needed.mid-point of the distribution is needed.

When extreme scores are likely to When extreme scores are likely to remarkably to affect the mean. The median remarkably to affect the mean. The median remains unchanged by the extreme scoresremains unchanged by the extreme scores

When it is desired to know the position of When it is desired to know the position of an individual score in terms of its an individual score in terms of its percentage distance from the mid-point of percentage distance from the mid-point of the distribution. the distribution.

USE OF THE MODEUSE OF THE MODE

When a rough and instant measure When a rough and instant measure of the central point of the distribution of the central point of the distribution is needed.is needed.

When the measure of the central When the measure of the central point is required to denote the most point is required to denote the most typical value or character of the typical value or character of the group.group.