MEASURES OF CENTRAL TENDENCY : MEASURES OF CENTRAL TENDENCY PRESENTATION BY:
DR. SAMPDA RAJURKAR. AVERAGE : AVERAGE Condensation of data in to single value mostly it is at centre & it carries important properties of data. Also known as MEASURES OF LOCATION or CENTRING CONSTANT. Different measures of central tendency : Different measures of central tendency 1. Mean : 1. Arithmetic mean
2. Harmonic mean
3. Geometric mean 4. Weighted mean
2. Median :
5. Average: Desirable properties of central tendency : Desirable properties of central tendency Should be rigidly defined.
Computation should be based on all observations.
Should lend itself for algebric treatment.
Should be least affected by extreme observations . Arithmetic mean : Arithmetic mean It is commonly used measure of central tendency.
It is sum all observations divided by number of observations For ungrouped data : For ungrouped data Mean of ‘n’ observations x1,x2…….xn is given by
A.M = X1+X2+…….+Xn
= sum of observations
Number of observations Example: : Example: 61, 58, 62, 67, 65, 68, 70, 69.
= 65 Short cut method / Assume mean method : Short cut method / Assume mean method When observations in data are large in size, it is laborious work to find mean. To avoid this it is used.
Assume arbitrary mean i.e. any value from data (a). Short cut method / Assume mean method : Short cut method / Assume mean method Subtract this assumed mean from each observation . We get what is k/a differences/deviations (d).
Obtain mean for deviations by usual method.
Mean for original data obtained by adding mean of deviations to assumed mean. FORMULA AND EXAMPLE: : FORMULA AND EXAMPLE: CONTINUOUS FREQUENCY DISTRIBUTION: : CONTINUOUS FREQUENCY DISTRIBUTION: As we know, frequency distribution, the frequency is not associated with any specified single value but spread over entire class. It creates difficulty for finding values x1, x2,…..xn. To overcome this difficulty we make reasonable assumption that the frequency is associated with mid values of class or the frequency is distributed uniformly over the class. Slide 12: From this we assume mid values as x1, x2…..xn. Of intervals and calculate arithmetic mean
x= sum fx
f METHOD: : METHOD: Write all class interval in 1st column and corresponding frequency in 2nd column.
Mid values of Class interval = lower+ upper cl.interval
which is put in 3rd column.
Multiply each “f” by corresponding “x” and write this product in 4th column.
Addition of this column gives “fx”. EXAMPLE: : EXAMPLE: SHORT CUT METHOD: : SHORT CUT METHOD: If values of variables are large in size, finding the mean is laborious. In that
1: any value from data is chosen called as assumed mean (a) SHORT CUT METHOD: : SHORT CUT METHOD: 2: take difference of assumed mean and mid values k/as deviation or differences (d)
3: multiply each “d” by corresponding “f”
4: calculate “bar d” by using formula
5: now original mean “bar x = a+ bar d” EXAMPLE: : EXAMPLE: MERITS OF A.M: : MERITS OF A.M: Though it seems to be best measure of central tendency it has certain limitations.
1: it is easy to calculate and understand.
2: it is based on all observations.
3: it is familiar to common man and rigidly defined.
4: it is capable of further mathematical treatment.
5: it is least affected by sampling fluctuations hence more stable. DEMERITS OF A.M: : DEMERITS OF A.M: 1:Used only for quantitative data not for qualitative data like caste, religion, sex.
2:Unduly affected by extreme observation.
3: Can’t be used open ended frequency distribution.
4:sometimes A.M may not be an observation in data.
5:Can’t be determined graphically. MEAN OF COMBINED GROUP: : MEAN OF COMBINED GROUP: Sometimes it is necessary to compute mean of two groups combined together.
If sizes & groups are known
Xc = n1x1+n2x2
n1+n2 GEOMETRIC MEAN(GM) : GEOMETRIC MEAN(GM) When data contains few extremely large or small values in such case arithmetic mean is unsuitable for data
GM of n observation is defined as ‘n’th root of the product of n observation HARMONIC MEAN: : HARMONIC MEAN: It is reciprocal of arithmetic mean of reciprocal observations. WEIGHTED MEAN : : WEIGHTED MEAN : We are considering that each item in data is of equal importance. Sometimes , this is not true, some item is more important than others. In such cases the usual mean is not good representative of data. Therefore we are obtaining weighted mean by assigning weights to each item according to their importance.
Ex…………… MEDIAN: : MEDIAN: Def: when all the observation of a variable are arranged in either ascending or descending order the middle observation is k/a median.
It divides whole data into equal portion. In other words 50% observations will be smaller than the median and 50% will be larger than it. COMPUTATION: : COMPUTATION: Ungrouped data…………….. Grouped data: : Grouped data: Obtain class boundaries.
Find less than cumulative frequencies of all the classes in data.
Compute N/2 and compare this quantity with less than cumulative frequencies. Choose less than cumulative frequency (CF) which is equal to or just exceeds the N/2. the class corresponding to this less than CF is k/a median class in which median lies. Grouped data: : Grouped data: Apply the formula and find mean
median = I+N/2-CFxh
f Grouped data: : Grouped data: I = lower boundary of median class
N = total frequency
C.F = less than cumulative frequency of the class previous to the median class
f = frequency of median class
h= class width Grouped data: : Grouped data: Wt in kg
4.5&above GRAPHICAL METHOD: : GRAPHICAL METHOD: The median can be obtained graphically from the ogive curve. For this, plot “ less than” for the given frequency distribution. Calculate the value of N/2 and locate it on “Y” axis. Draw a line from this point which is parallel to “X” axis to meet the ogive curve. From the point of intersection drop perpendicular on X axis
Here median = value where perpendicular cuts X axis. MERITS: : MERITS: Easy to understand & calculate.
It can be computed for a distribution with open end classes.
It is not affected d/t extreme observation.
Applicable for qualitative and quantitative data.
Can be determined graphically. DEMERITS: : DEMERITS: It is not based on all observations, hence it is not proper representative.
Not rigidly defined as A.M.
Not capable of further mathematical treatment. MODE: : MODE: The observation which occurs most frequently in a series is k/a MODE. Ungrouped data: : Ungrouped data: Mode is obtained by inspection. GROUPED DATA: : GROUPED DATA: Obtain class boundaries.
Locate the model class is the class which has maximum frequency.
Find mode by using formula.
Mode =I+Fm-F1 h
2Fm-F1-F2 GROUPED DATA: : GROUPED DATA: Where,
I=lower boundary of modal class
Fm =frequency of modal class
F1=frequency of pre modal class
F2=frequency of post modal class
h=width of modal class Example: : Example: ……………..
Procedure applicable to unimodal distribution only.
Mode can’t be determined if modal class is at the extreme. GRAPHICAL METHOD: : GRAPHICAL METHOD: Graphical demonstration can be made by plotting histogram. MERITS: : MERITS: As compared with mean & median mode has very limited utility
It is applicable for qualitative & quantitative type of data.
It is not affected by extreme observations.
It can be determined even though distribution has open end classes.
It can be obtain graphically. EMPERICAL RELATION: : EMPERICAL RELATION: MEAN-MODE=3(MEAN-MEDIAN) PARTITION VALUES: : PARTITION VALUES: The values which divide the given data in to number of equal parts are called the partition values.
The most commonly used partition values are QUARTILES, QUINTILES, DECILES. QUARTILES: : QUARTILES: The values which divide the given data in to four equal parts when observations are arranged in order of magnitude are k/a quartiles.
obviously there will be three quartiles Q1,Q2 & Q3.
Q1(1st quartile):25%below &75%above
Q2(2nd quartile): same as median 50% above & below
Q3(3rd quartile):75%below &25% above QUINTILES & DECILES: : QUINTILES & DECILES: Quintiles : It contains four points so it will divide data in to five equal parts.
Deciles : it contain 9 points & it will divide data in to ten equal parts. Slide 44: HAVE
DAY! THANK YOU!