Statistics and Market Research

Population data help the businessman considerably in exploring new markets for his product; they serve as guideposts to market demand. The information on the consumer’s preference, buying habits, levels of living and income, together with their competition to be met, and the cost of operating business should be carefully studied.

It should be noted that not all commodities enjoy large sales even in communities that are thickly populated. Hence, a dealer in working clothes should take into account the number of farmers and laborers in the community, not the total population. Dealers in tractors, in farm machineries, and in agricultural implements may use agricultural income as a good index in determining their sales potentials. Dealers in gasoline should consider the number of motor vehicles in the place of business as a good measure of probable volume of sales.

Analysis of Variance (ANOVA)

The analysis of variance (ANOVA) is the statistical method used for testing the null hypothesis that the means of several populations are equal. The comparison in means of 3 or more populations which follow normal distributions can be taken simultaneously in just one application of this test. This test, therefore, is a generalization of the z and t tests of two normal population means.

In order to use ANOVA, certain conditions must be met:

1. The sample(s) must be randomly selected from normal populations.
2. The populations should have equal variances.
3. The distance from one value to its group’s mean should be independent of the distances of other values to that mean (independence of error).

ANOVA is reasonably robust, so that minor variations from normality and equal variance are tolerable.
ANOVA uses squared deviations or variances so that computation of distances of individual data points from their own mean or from the grand mean can be summed.

The test statistic for ANOVA is the F ratio. It compares the variance from the two sources:

F = Between – groups variance / Within – groups variance
F = Mean Square between (MSB) / Mean Square within (MSW)
F = MSB / MSW

Where:
MSB = Sum of Squares between / Degrees of Freedom between
MSB = (SSB) / (dfB)
MSW = Sum of Square within / Degrees of Freedom within
MSW = (SSW) / (dfw)
The degree of freedom for SSB = k – 1 and for SSW = k (n – 1)

Where:
k = number of groups or samples
n = no. of items per column (size of each sample)

Correlation Formula

CORRELATION= is a measure of relationship between two variables.
Coefficient of correlation determine validity, reliability and objectivity of an examination prepared. It also indicates the amount of agreement or disagreement between groups of scores, measurements, or individuals.
Interpretation of Ranges
+ 0.00 to + 0.20 –Slight correlation, almost negligible relationship
+ 0.21 to + 0.40 –Slight correlation, definite but small relationship
+ 0.41 to + 0.70 –moderate correlation, substantial relationship
+ 0.71 to + 0.90 –High correlation, marked relationship
+ 0.91 to + 1.00 –Very high correlation, very dependable relationship
Coefficient of correlation Spearman’s Formula:
R=1-[(6(ΣG)/N2-1]
Where:
G=Difference of the two ranked scores
N=Number of scores
Procedure:
1. Write the scores or measures of the two variable under column x and column y
2. Rank the scores under column x , with the highest score as rank 1 and the lowest score as rank N. Write the ranks of the scores under column Rx which means rank of x
3. Rank the scores under column y with the highest scores as rank 1 and the lowest score as rank N.
4. Subtract the Ry values from the Rx values. Write the difference under column G, means gain. Consider only the positive values.
Coefficient of correlation by the use of the Rank-Difference Method:
rho=1-[(6(ΣD2)/N(N2-1)]
Procedure:
1. Follow the same steps from 1 to 3 in the Spearman’s Formula.
2. Find the difference between the two steps of ranks or values under column Rx and Ry. Subtract the larger value from the smaller value.
3. Write the difference of Rx and Ry under column D, which means difference.
4. Square the difference, D and write under column D2.
5. Get the sum of the values under D2.
Coefficient of correlation by the Product-Moment Method:
rxy=Σdxdy/square root[(Σd2x)(Σd2y)]
Procedure:
1. Get the total of the data under test x and test y and find the mean of x and mean of y.
2. Get the deviations dx and dy by getting the difference between the mean and the scores.
3. Square dx to obtain d2x and dy to obtain d2y.
4. Get the summation of each.
5. Get the product of dx and dy to have dxdy
6. Get the summation of dxdy.

Set Operations

1. With any two sets “A” and “B” there is associated a third set “C” satisfying the property that C = { X/X Є AV x Є B }

In words: “C” is equal to X, such that X is belong to “A” or X is an element of B

“C” is called the union of “A” and “B” we denote the set C symbolically as C = A U B

Example:
A = {3, 4, 5, 6, 7}
B = {2, 4, 6, 8, 10}
A U B = {2, 3, 4, 5, 6, 7, 8, 10}

2. With any two sets A & B there is associated A third set “D” satisfying the property that D = { X/X Є A ۸ X Є B}

In words: D equals X such that X is an element of set “A” and x is a member of B.

“D” is called the intersection of sets A and B, and we denote the set D symbolically as D = A B

3. With any two sets A and B there is associated A third set “C” satisfying the property that C = { X/X Є A ۸ X € B}. We denote the set symbolically as C = A – B, and call C the relative complement or difference of A and B.

Example:
A = { a, b, c, d, e, f}
B = { a, e, i, o, u}
A – B = {b, c, d, f} and B – A = {i, o, u}

4. If A is a subset of U, then the set of an elements contained in U that are not elements of A is called the complement of A in U and is designated by Ă then Ă = {X/X Є U ۸ X € A}

Example: Consider the universal set of an counting nos. and the set A of counting numbers less than 100 then

U = {1, 2, 3, 4, ……} A = { 1, 2, 3,……99}

Ă = {100, 101, 103…….}

5. The set product or cartesian product of two sets A and B is the set of an possible ordered pairs (a, b) where a is in A and b is in B. We symbolize this set of ordered pairs by A X B and write,
A X B = {(a,b) / a Є A ۸ b Є B }

Example:

If A = {1, 2} and B = {x, y} then A X B = { (1, x), (1, y), (2, x) , (2, y}} and B X A = {(X, 1), (X, 2), (Y, 1), (Y, 2)}

Kinds of set

1. Finite set – countable
Example: Sets A, B, C, D are finite sets

2. Infinite set – uncountable
Example: Set E is an infinite set

3. Empty or null set – has no element
Example: A = { }

4. Equal set – set A and set B are equal set if the elements of set A is exactly the element of set B.
Example:
A = {set of an even counting number of one digit} = {2,4,6,8}
B = {set of an integral multiples of two having one digit = {2,4,6,8}

5. Equivalent set – two sets are equivalent if there exists a one-to-one correspondence between elements of the two sets.
Example:
A = {1, 2, 3, 4,5} - x coordinate
B = {6, 7, 8, 9, 10} – y coordinate

then “A” is equivalent to B. We can construct the relation of set A and set B.

{ (1,6}, (2,7), (3,8), (4,4), (5,10) }

6. Subset – set whose elements are members of the given set A = {1,2,3,4,5,8}, B = {2,4,8}

7. Universal Set – totality of the given set with consideration. The set from which we select elements to form A given set is called universal.
Example:
Set A = {1, 2, 3, 4, 5, 8} is a universal set
Set B = {2, 4, 8} is a subset of set A

8. Disjoint Set – sets that has no common element ; if two sets have no element in common, the sets are called disjoint sets.

Methods of Writing Set

Example: Roster Method and Rule Method

Methods of Writing Set

1. Roster or tabular method

The elements of the set are enumerated and separated by comma.

2. Rule method or set builder

A, descriptive phrase is used to describe the elements of the set

Sets Definition and Examples

Set

Definition:
Set is a well-defined collection of things or objects

Note:
Sets maybe denoted by capital letters such as A,B,X, Y

An element or member of a set is a thing that belongs to the set and maybe denoted by small letters such as a,b,c……..x,y.

The members of the set are enclose in braces { }, with a comma separating the members.

Example:

The set “A” whose members are ETHEL, CYNTHIA, CHELO, we usually use the symbol.

A = {ETHEL, CYNTHIA, CHELO}

ETHEL Є A
- Read as ETHEL is an element of set A
- Read as ETHEL belongs to set A
- Read as ETHEL is a member of set A