Statistics: Mean

Mean Math Or Mean Statistics
In Mathematics in the branch of Statistics, the expression for the mathematical mean of a statistical distribution is the mathematical average of all the terms in the data. To calculate this, we add up the values of the terms given and divide the sum by the number of terms in the data. This expression is also called the Arithmetic Mean.
Example: Find the Arithmetic Mean of the following data 5, 5,10,10,15,15,20,30
Solution: Arithmetic Mean = Regular average = sum of the values of the terms/number of terms
Sum of the values of the terms =5+5+10+10+15+15+20+30=110
Number of terms = 8
Mean = 110/8 = 13.75
Sample Mean
The sample mean in statistics branch of Mathematics is the sum of all observed outcomes from the sample divided by the total number of events. It is denoted by the symbol x with a (bar) above it. The formula used to compute the sample mean is as follows:
X (bar)= (1/n) (x1+x2+x3………xn)
 If we consider the sampling of some non-indigenous weed in a land of five acres in Springfield and came up with the counts of this weeds in that area as 38, 56, 84,105,116
We calculate sample mean for the above sampling by adding the weed counts and divide by the number of samples, 5
(38+56+84+105+116)/5= 79.8
So, we get the sample mean of non-indigenous weeds = 79.8
Weighted Mean
While computing Arithmetic Mean all the terms or values have equal importance. But at times we come across some situations where all the terms or values do not have equal importance. For instance, when we compute the average number of marks per subject, as per the different subjects like English, Science, Mathematics, Social Sciences; each subject has different levels of importance.
So, weighted mean is the arithmetic mean calculated by considering the relative importance (weight) of each term.
Each item is assigned a weight in proportion to its relative importance
Formula to compute Weighted Mean: xw (bar) = sigma(wx)/sigma(w)
(here x =value of the items  w= weight of the item)
Example:  A student scored 50, 70, 80 in English, Science and Math respectively and assume the weights of each subject to be 4,5,6  respectively. Find the weighted arithmetic mean of each subject.
Solution: let us tabulate the given data
Subjects              Marks obtained          weight              wx
English                50                          4                  150
Science                       60                          5                  300
Math    80                          6                  480
         Sigma(w)=15  sigma(wx)=930
Arithmetic Weighted mean = 930/15= 62 marks/subject
Short-cut Method
A short-cut method of calculating the arithmetic mean is based on the property of arithmetic average. In this method the deviations (D) of the items from an assumed mean are first calculated and then multiplied with their respective frequencies (f). Then, the total of these products [sigma(fD)] is divided by the total frequencies [sigma(f)]and added to the assumed mean(A). The figure we get is the actual arithmetic average or the Arithmetic Mean.
Formula used in the short-cut method of calculating the arithmetic mean:
X(bar) = A + sigma(fD)/sigma(f)

Operations on Decimals

Decimals:
Numbers representing fractions without having any numerator or denominator is called a decimal number.
Example: 0.567, 0.2, 0.49
Decimal Chart:
The value of a number in a decimal is decided by its place value.
0.345
3 represents three tenths = 3/10
4 represents four hundredths = 4/100
5 represents five thousandths = 5/1000

In engineering calculations, fractions are used extensively. As we know that, we can always express a fraction in terms of its corresponding decimal number. We have a ready reckoner that gives us corresponding converted values of a fraction.
1/64   = .015625 1/32   = .03125 3/64  = .046875 1/16 = .0625
5/64   = .078125 3/32   = .09375 7/64  = .109375 1/8  = .125
9/64   = .140625 5/32   = .15625 11/64 = .171875 3/16 = .1875
13/64   = .203125 7/32   = .21875 15/64 = .234375 1/4  = .25
17/64   = .265625 9/32   = .28125 19/64 = .296875 5/16 = .3125
21/64   = .328125 11/32  = .34375 23/64 = .359375 3/8  = .375
25/64   = .390625 13/32  = .40625 27/64 = .421875 7/16 = .4375
29/64   = .453125 15/32  = .46875 31/64 = .484375 1/2  = .5
33/64   = .515625 17/32  = .53125 35/64 = .546875 9/16 = .5625
37/64   = .578125 19/32  = .59375 39/64 = .609375 5/8  = .625
41/64   = .640625 21/32  = .65625 43/64 = .671875 11/16= .6875
45/64   = .703125 23/32  = .71875 47/64 = .734375 3/4  = .75
49/64   = .765625 25/32  = .78125 51/64 = .796875 13/16= .8125
53/64   = .828125 27/32  = .84375 55/64 = .859375 7/8  = .875
57/64   = .890625 29/32  = .90625 59/64 = .921875 15/16= .9375
61/64   = .953125 31/32  = .96875 63/64 = .984375

Multiplying decimals
In multiplication of decimal numbers, we follow two steps:
Step 1: Multiply the numbers without taking into consideration of decimal points
Step 2: Place the decimal point starting from right and moving as many digits as the sum of the number of digits in decimal part in both numbers towards left.
Example:


Dividing decimals:
We have two steps to follow.
Step 1: Convert both numerator and denominator into fractions
Step 2:  Follow the rules of division of fractions.
Example: 14.4 ?0.12
Step 1: 14.4 = 144/10 and 0.12 = 12/100
Step 2: 144/ 10 ? 12/100 = 144/10 ?100/12 = 120
Subtracting decimals:
To subtract decimals one from the other we follow
Step 1: Write the numbers one below the other with decimal point coinciding
Step 2: Add zeros to make length of the numbers same
Step 3: Follow the laws of subtraction to subtract
Example:
0.56- 0.287
0.560
0.287
-------
0.213
Terminating decimals:
The word terminate means end. A decimal number with definite number of digits (ending) is called terminating decimal.
Example: 0.495. There are three digits after the decimal point.
All terminating decimals can be represented as a rational number.
Dewey Decimal System:
In public libraries, books are catalogued according to the subjects. Dewey Decimal System (DDC) is the most widely used system of classification for organizing books in libraries. In this system, integer part of the number gives us the specific subject and decimal part gives us the specific part of the subject.
For example:
In DDC, 512.12 represents a book on Algebra, 815.4 represents a book on 16th century Italian poetry.

Exponents and their rules

Fractional exponent is also called as a rational exponent which is in the form of a fraction like ½ (square root), ⅓ (cube root), ¼ (fourth root) etc.

A fractional exponent with 1/n is the n-th root of the given base.

(a)^ ⅓ = ³√a

A fractional exponent with m/n, we need to take the m-th power of the base and then find the n-th root or vice versa

(a) ^⅔ = ³√a²

The basic exponential rules are as follows: 

Exponents and tLaw of exponents:

1. Any number except zero when raised to zero equals 1
2. Any number raised to 1 equals itself
3. A number with negative power equals its reciprocal with a positive power
4. When we divide terms with same bases, the powers are subtracted
5. When we multiply terms with same bases, the powers are added
6. When a term is raised to a power with a whole power, the powers are multiplied.
7. When product of terms are raised to a power, each term is raised to that power

Subtracting exponents:

When we divide terms with same bases, the exponents are subtracted

The basic exponential rules are as follows: 

For example,

= 3³

Exponent rules:

1. When we multiply terms with same bases, the powers are added

2. When we divide terms with same bases, the powers are subtracted

3. Any number except zero when raised to zero equals 1

4. Any number raised to 1 equals itself

5. When a term is raised to a power with a whole power, the powers are multiplied.

6. When product of terms are raised to a power, each term is raised to that power

7. A number with negative power equals its reciprocal with a positive power