MEASURES OF CENTRAL TENDENCY:
These are used to measure the 'central' or 'expected' value of the data set.
Standard deviation and Variance are measures of spread. That is, they indicate how spread the data is.
Five most common measures of central tendency:
1. Arithmetic mean
2. Geometric mean
3. Harmonic mean
4. Median
5. Mode
CONCEPT 4
ARITHMETIC MEAN:
It is the method of calculating standard average, also called as 'mean'. It is used to find the central tendency of any random distribution of numbers.
A.M = (x1+x2+x3+.... +xn) / n
Example:
Company A has shown profit of Rs.30 lakhs, 42, 45, 48 and 52 for rhe last five years
Company B has shown profit of Rs. 28 lakhs, 46, 50, and 57 for the last four years
Which company has shown better profits?
This can be answered by calculating the arithmetic mean of both companies A and B
A.M of A = 43.4 lakhs A.M of B = 45. 25 lakhs => Obviously, Company B had higher profits.
NOTE
1. AM is not a very good measure of central tendency.
Why? the mean tends to be influenced by highest value of the set. Example: Average marks of a set of 10, 12, 12, 12, 13, and 25=> AM= 14 But, 5 out of 6 scores are below than this. Median should be adopted in such cases.
CONCEPT 5
GEOMETRIC MEAN:
G.M of n positive numbers is the mean calculated by taking nth root of the product of these numbers.
GM = (x1*x2*x3*.....*xn)^ (1/n)
Geometric mean is used to calculate the average growth rates or interest rates.
Example 1:
A certain amount is deposited in a bank. For four years, the bank compounds the interest rate of 10%, 12%, 14% and 15% respectively. Compounded net interest rate = ?
Soln:
If the principal is Rs.100, the amount at the end of first year, P1 = 100* 1.1
Likewise, After 4 years, P4 = 100*1.1*1.12*1.14*1.15 ---- (1)
Another way of expressing P4 is by compound interest formula
P4 = 100 *(1+(r/100))^4--- (2) Equating 1 and 2, we get r = 12.7 %
Example 2:
If the population of a village increases by 10 % in one year and then decreases by 20 % in next year,
The average rate of increase/decrease =? This type of problem can be solved by GM of the percentages.
Soln:
First year increase of 10% =>The value will be 100+10 = 110%
Next year decrease of 20% => The value will be 100-20 = 80%
So, the average rate will be (110%*80*)^(1/2) which is the GM = 93.8%
Therefore, the average rate of decrease per year = 6.2%
Check: Assume a population of 100 people. In first year, 10% increase will give 110. In second year, 20% decrease of 110 will give 88. => So, there has been 12% decrease in two years.
Approximately, 6% decrease per year.
Example 3:
You need to take vehicle loan from a bank. The fixed rate is 12% for the last 3 years. The floating rates are 10%,12% and 13% for the last 3 years. Which one is best to adopt.
Soln:
Fixed rate is same = 12%
GM of floating rate = (1.10*1.12*1.13)^(1/3) = 1.116 => Avg floating interest for 3 years = 11.6%
So, adoption of floating rate of interest is best.
CONCEPT 6
HARMONIC MEAN:
Harmonic mean of a set of numbers is given by the following formula.
HM = n / ((1/x1)+(1/x2)+(1/x3)+....(1/xn))
For any two quantities a and b, HM is given by the formula,
HM = 2ab / a+b
NOTE:
1. Harmonic mean is the least of the three Pythagorean means (AM, GM and HM). Why? The HM tends towards the smaller values which is opposite to AM which tends towards larger values
2. HM is used in many situations involving rates and ratios. In such cases, HM gives the best average.
3. Also, HM is used to find average speed when equal distances are covered at different speeds
Example 1:
A person travels from city P to city Q of distance 300 km at an average speed 60 km/hr.
On the way back, from Q to P, he covers the same distance at 50 km/hr. Average speed for entire journey=?
Soln:
Average speed is the HM of the two numbers,
Average speed = (2*60*50)/(50+60) = (6000/110) = 54.54 km/hr
ALTERNATE METHOD TO CALCULATE AVERAGE SPEED
A person plans a trip from city A to city B
Decides to travel half the distance by flight at speed of 336 km/hr
And half of the remaining distance by train at 60 km/hr
And the rest by car at 40 km/hr What is the estimated average speed?
Soln:
Notice here, the distances travelled by train and car are equal, now first calculate the average speed of train and car. Step 1=> Ratio of smaller speed to larger speed => 40:60= 2:3 (x:y)
Step 2=> Difference between two speeds |40-60|= 20 kmph
Step 3=> Difference/ (x+y) => 20/5 = 4
Step 4 =>Value obtained by step 3 *x => 4*2 = 8
Step 5=>Add the value obtained in step 4 to smaller speed => 40+8 = 48 kmph
To find the average speed of the trip, 48:336 => 1: 7
Difference = 336-48= 288, Average speed = 48+(288/8)*1 = 84 kmph
CONCEPT 6
RELATION BETWEEN MEANS:
For two numbers a and b,
1. AM = (a+b)/2
2. GM= (a*b)^(1/2)
3. HM = 2ab/(a+b)
4. GM = sqrt (AM*HM)
5. AM, GM and HM will be in geometric progression, AM>=GM>=HM. This will be useful in inequalities.
CONCEPT 7
WEIGHTED AVERAGES:
The term 'weight' stands for the relative importance that is attached to the values For example, consider the scenario that the average ages of different departments in a company are known, and there is a need to calcuate the combined average. In such case, a simple mean would not be correct because the 'weight' (no of persons) in different departments will vary. So, the weighted average is given by
Weighted average = (w1x1 +w2x2 +w3x3+...+wnxn) / (x1+x2+x3+...+xn) where w1, w2 and w3 are the weights of the respected values.
Example 1:
Weighted average of 1L milk @ Rs. 30, 4L milk @ Rs. 22 , 2L milk @ Rs.25 =?
Weighted average = ((1*30)+(4*22)+(2*25))/ (1+4+2) = Rs. 24
Example 2:
Three math classes X, Y and Z take an algebra test.
Average score of X is 83. Avg score of class Y is 76. Average score of class Z is 85
Average score of class X and class Y is 79. Average score of class Y and class Z is 81.
What is the average score of all the three classes? (CAT 2001)
Soln:
Let no of students in classes X,Y and Z be x, y and z respectively
=> 63x+76y = 79( x+y) --- (1) ; 76y+85z = 81(y+z) ---- (2)
=> 4x= 3y ; 4z= 5y
Combining, 4x= 3y ; 4z= 5y by the rule of proportions
( given that a:b and b:c, To find a:b:c => a*b: b*b: b*c), we get 20x = 15y = 12 z
which is, x:y:z = 3:4:5
The average of three classes= (83*3)+(76*4)+(85*5) /(3+4+5) = 81.5
Example 3:
A final year MBA student gets 50% in exam and 80% in assignments. If the exam should count for 70% of final result and 30% for assignment, what will be final score if the college decide to use weighted harmonic mean to uneven performances? (FMS 2007)
Weighted harmonic mean = (W1+W2)/((W1/X1)+(W2/X2))
= (70+30)/((70/50)+(30/80)) = 56.34% (app)
CONCEPT 7
MEDIAN:
Median is the middle value of the group of numbers arranged in an ascending order or descending order.
1.If the number of values in a given set is odd, then the median will be (n+1)/2 value
Median of 31, 43, 32, 34, 36, 42, 33, 39 and 40.
First arrange in ascending/descending order => 31, 32, 33, 36, 39, 40, 42, 43 and 45.
No of values = 9 (odd), median = (9+1)/2 = 5th value => 39
2. If the number of values in a given set is even, then there will be two middle values say x and y. Median will be (x+y)//2.
Median of 102, 99, 111, 101, 98, 87, 105, 100.
First arrange in ascending/descending order => 111, 105, 102, 101, 100, 99, 98, 87
No of values = 8(even), median = (8/2) = average of 4th and 5th value => (101+100)/2 = 100.5
NOTE:
The median of a set of values may or may not be equal to, less than or greater than the mean set of values.
Example 1:
In the data set {2,5,7,8,X}, the arithmetic mean is the same as median. Determine the value of X. Assume X>=8 (JMET 2009)
Soln:
Given that the median is same as Arithmetic mean. So, calculate the median first.
n= 5. Median = (5+1)/2 = 3rd value = 7 => Therefore,Arithmetic mean =7
That is (2+5+7+8+X)/5 = 7 => X =13
CONCEPT 8
MODE:
1. Mode is the number that occurs most frequently in a given set of numbers.
2. If two or more values appear the same number of times, then the data set does not have a unique mode.
3. The mode of a set may or may not be equal to the median or mean of the set of values.
Example 1:
Find the mode of the set 1,2,3,7,9,2,12,2,13,4,10,2,8,2,6 ?
Arranging in ascending order => 1,2,2,2,2,2,3,4,5,6,7,8,9,10,12,13
2 occurs maximum number of times ( 5 times) So, the mode is 2
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