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
Monday, December 29, 2008
Wednesday, December 17, 2008
Empty Set and Set
- A set is a collection of things
- An element or member of a set is a thing that belongs to the set.
* There are many words which we use in everyday language that have the same meaning as the word set.
Example:
1. A herb of cattle is a set of cattle
2. A flock of birds is a set of birds
3. A squadron of planes is a set of planes
4. a school of fish is a set of fish
5. A regiment of soldiers is a set of soldiers
* The members of the set are enclosed in braces, { }, with a comma separation the members and to identify sets we often name them by capital letters.
Example:
1. The Set “A” whose members are Ethel, Emerson and Merecel. We usually use the symbol
A = {Ethel, Emerson, Merecel}
2. The Set “B” of days of the week
B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
The set C of words to distinguish two faces of a coin
C = {Heads, Tails}
* The set that has no elements is called the empty set, we use the symbol Ǿ to indicate the empty set.
Example of Empty set:
1. the set of whole numbers by 9 and 10.
2. the set of four-sided triangles.
3. the set of astronauts who have landed on the planet Pluto
4. the set of icebergs in the sahara desert
5. the set of people with two heads
6. the set of pink elephants
7. the set of purple cows
- An element or member of a set is a thing that belongs to the set.
* There are many words which we use in everyday language that have the same meaning as the word set.
Example:
1. A herb of cattle is a set of cattle
2. A flock of birds is a set of birds
3. A squadron of planes is a set of planes
4. a school of fish is a set of fish
5. A regiment of soldiers is a set of soldiers
* The members of the set are enclosed in braces, { }, with a comma separation the members and to identify sets we often name them by capital letters.
Example:
1. The Set “A” whose members are Ethel, Emerson and Merecel. We usually use the symbol
A = {Ethel, Emerson, Merecel}
2. The Set “B” of days of the week
B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
The set C of words to distinguish two faces of a coin
C = {Heads, Tails}
* The set that has no elements is called the empty set, we use the symbol Ǿ to indicate the empty set.
Example of Empty set:
1. the set of whole numbers by 9 and 10.
2. the set of four-sided triangles.
3. the set of astronauts who have landed on the planet Pluto
4. the set of icebergs in the sahara desert
5. the set of people with two heads
6. the set of pink elephants
7. the set of purple cows
Tuesday, December 16, 2008
Permutation Formula and Example
Permutation
Each different arrangement or ordered set of object is caused a permutation of those objects.
- if A = {a1, a2, a3……. An} is any set of n elements then any arrangement of the elements of “A” along a line is called a permutation of the elements of A.
All the permutation of the elements of the set is given by the formula:
P = n! where n = no. of elements
Problem:
How many permutations can be made from the word PINOY”
Solution:
PINOY – consist of 5 letters
P = 5! = 120 permutations
The total no. of permutations of n objects taken r at a time, P(n,r) is given by the expression.
P (n,r) = nPr = n!/(n-r)!
Problem:
Find the no. of permutations of the four integers 1,2,3,4 taken two at a time.
Solution:
n = 4, r = 2
4P2 = n!/(n-r)! = 4!/(4-2)! = 4!/2! = 4.3.2.1 / 2.1 = 12
Each different arrangement or ordered set of object is caused a permutation of those objects.
- if A = {a1, a2, a3……. An} is any set of n elements then any arrangement of the elements of “A” along a line is called a permutation of the elements of A.
All the permutation of the elements of the set is given by the formula:
P = n! where n = no. of elements
Problem:
How many permutations can be made from the word PINOY”
Solution:
PINOY – consist of 5 letters
P = 5! = 120 permutations
The total no. of permutations of n objects taken r at a time, P(n,r) is given by the expression.
P (n,r) = nPr = n!/(n-r)!
Problem:
Find the no. of permutations of the four integers 1,2,3,4 taken two at a time.
Solution:
n = 4, r = 2
4P2 = n!/(n-r)! = 4!/(4-2)! = 4!/2! = 4.3.2.1 / 2.1 = 12
Monday, December 15, 2008
Statistics Probability Sample Problems
1. At a certain canteen, Doris can choose merienda from three drinks (Coke, Pepsi, Gulaman) and four sandwiches from (bacon, chicken, tuna, egg). In how many ways.
Solution:
D = {Coke, Pepsi, Gulaman}
N(D) = 3
S = {Bacon, Chicken, Tuna, Egg}
N(S) = 4
N1 . N2 = 3 x 4 = 12 ways
2. Two dice are rowed, in how many ways can they fall? If 3 dice are rowed? and if 4 dice are rowed?
For two dice
N1 = 6
N2 = 6
N1.N2 = 6 x 6 = 36 ways
For three dice
N1 . N2 . N3
6 x 6 x 6 = 216 ways
For four dice
N1.N2.N3.N4
6 x 6 x 6 x 6 = 296 ways
3. Using the digits 1,2,3,4,5,6, How many two-digit can be formed if a) repetition is allowed b) repetition is not allowed. How many numbers do we have to choose from the given set, they are 6 numbers.
Solution:
a) Repetition is allowed
6 x 6 = 36 ways
b) Repetition is not allowed
6 x 5 = 30 ways
Solution:
D = {Coke, Pepsi, Gulaman}
N(D) = 3
S = {Bacon, Chicken, Tuna, Egg}
N(S) = 4
N1 . N2 = 3 x 4 = 12 ways
2. Two dice are rowed, in how many ways can they fall? If 3 dice are rowed? and if 4 dice are rowed?
For two dice
N1 = 6
N2 = 6
N1.N2 = 6 x 6 = 36 ways
For three dice
N1 . N2 . N3
6 x 6 x 6 = 216 ways
For four dice
N1.N2.N3.N4
6 x 6 x 6 x 6 = 296 ways
3. Using the digits 1,2,3,4,5,6, How many two-digit can be formed if a) repetition is allowed b) repetition is not allowed. How many numbers do we have to choose from the given set, they are 6 numbers.
Solution:
a) Repetition is allowed
6 x 6 = 36 ways
b) Repetition is not allowed
6 x 5 = 30 ways
Sunday, December 14, 2008
Statistics Probability: Definitions, Principles and Samples
Probability which connotes the “chance” or the “likelihood” that something will happen or occur is an interesting and fascinating area of mathematics.
Probability – the part of mathematics that deals with the questions “how likely” is called probability or the theory of probability.
Probability – is a measure of certainty, its scale is from 0 to 1. A probability of zero indicates that there is no chance at all that an event will happen or occur. A probability of one (1) indicates absolute certainty that an event will happen. Absolute certainly rarely happens in lifes.
1. Experiment
Activity that can be done repeatedly.
Examples:
1. Tossing a coin
2. Rolling a pie
2. Sample Space – set of all possible outcomes in an experiment(s)
Examples:
a.) S = {H,T} n(S) = 2
b.) S = {1,2,3,4,5,6} n(S_ = 6
c.) S = {Rod, Ed, Emer} n(S) = 3
3. Sample Point – an element in the sample space
Examples
a.) H is a sample point
T is a sample point
4. Event – is a subset of sample space
Example:
Getting an even number when you roll a die is an event
S = {1,2,3,4,5,6}
E = {2,4,6}
n (E) = 3
Counting Techniques
N1 . N2 . N3 . N4 …..Nn (where N = event)
Fundamental Principles
If one thing can be done independently in N1 different ways and if a second thing can be done independently in N2 different ways and so on. Then the total number of ways in which all the things may be done in the stated order is N1 . N2 . N3 . N4 ……….
Probability – the part of mathematics that deals with the questions “how likely” is called probability or the theory of probability.
Probability – is a measure of certainty, its scale is from 0 to 1. A probability of zero indicates that there is no chance at all that an event will happen or occur. A probability of one (1) indicates absolute certainty that an event will happen. Absolute certainly rarely happens in lifes.
1. Experiment
Activity that can be done repeatedly.
Examples:
1. Tossing a coin
2. Rolling a pie
2. Sample Space – set of all possible outcomes in an experiment(s)
Examples:
a.) S = {H,T} n(S) = 2
b.) S = {1,2,3,4,5,6} n(S_ = 6
c.) S = {Rod, Ed, Emer} n(S) = 3
3. Sample Point – an element in the sample space
Examples
a.) H is a sample point
T is a sample point
4. Event – is a subset of sample space
Example:
Getting an even number when you roll a die is an event
S = {1,2,3,4,5,6}
E = {2,4,6}
n (E) = 3
Counting Techniques
N1 . N2 . N3 . N4 …..Nn (where N = event)
Fundamental Principles
If one thing can be done independently in N1 different ways and if a second thing can be done independently in N2 different ways and so on. Then the total number of ways in which all the things may be done in the stated order is N1 . N2 . N3 . N4 ……….
Saturday, May 31, 2008
Mode
Mode = is that single measure or score which occurs most frequently. When data are grouped into a frequency distribution, the crude mode is usually taken to be the midpoint of that interval which contains the largest frequency.
When to use the mode:
1. When a quick and approximate measure of central tendency is all that is wanted.
2. When the measure of central tendency should be the most typical value.
Finding mode from the ungrouped data:
Example:
1. A set of numbers 11, 12, 13, 16, 16, 16, 19, 20 has 16 as the mode.
2. A set of numbers 45, 49, 52, 55, 58 has no mode.
3. A set of numbers 4, 4, 6, 8, 8, 8, 9, 9, 9, 10 has modes of 8 and 9 and is called bimodal.
Mode of grouped data
To determine the mode of grouped data we have to find first the modal class. In a frequency distribution, the modal class can be easily determine by inspection as it is the class with the highest frequency.
Mo = Lmo + [ d1/d1 + d2 ] c
Where: Lmo = lower boundery of the modal class
d1 = difference between the frequency of the modal class and the frequency of the class next lower in value.
d2 = difference between the frequency of the modal class and the frequency of the class next higher in value.
C = class size
Find the mode of example 3.8. table 3.1
Weekly wage ( in peso) f Lower class boundary
P 870-899 4 869.5
900-929 6 899.5
930-959 10 929.5
960-989 13 959.5
990-1019 8 989.5
1020-1049 7 1019.5
1050-1079 2 1049.5
Mo = 959.5 + [ 3/3+5 ] 30 = P 970.75
When to use the mode:
1. When a quick and approximate measure of central tendency is all that is wanted.
2. When the measure of central tendency should be the most typical value.
Finding mode from the ungrouped data:
Example:
1. A set of numbers 11, 12, 13, 16, 16, 16, 19, 20 has 16 as the mode.
2. A set of numbers 45, 49, 52, 55, 58 has no mode.
3. A set of numbers 4, 4, 6, 8, 8, 8, 9, 9, 9, 10 has modes of 8 and 9 and is called bimodal.
Mode of grouped data
To determine the mode of grouped data we have to find first the modal class. In a frequency distribution, the modal class can be easily determine by inspection as it is the class with the highest frequency.
Mo = Lmo + [ d1/d1 + d2 ] c
Where: Lmo = lower boundery of the modal class
d1 = difference between the frequency of the modal class and the frequency of the class next lower in value.
d2 = difference between the frequency of the modal class and the frequency of the class next higher in value.
C = class size
Find the mode of example 3.8. table 3.1
Weekly wage ( in peso) f Lower class boundary
P 870-899 4 869.5
900-929 6 899.5
930-959 10 929.5
960-989 13 959.5
990-1019 8 989.5
1020-1049 7 1019.5
1050-1079 2 1049.5
Mo = 959.5 + [ 3/3+5 ] 30 = P 970.75
Median
b.) Median = in the midpoint of the distribution. Half of the values in a distribution fall below median and the other half fall above it.
When to use the median:
1. When the exact midpoint f the distribution is wanted the, 50% point.
2. When there are extreme scores which would markedly affect the mean. Extreme scores do not disturb the median.
3. When it is desired that certain scores should influence the central tendency, but all that is known about them is that they are above or below the median.
- determine of whether the cases fall within the lower halve or the upper halve of a distribution (appropriate locator of central tendency).
Finding the median from ungrouped data:
1. When N is odd, the median is the middle score.
Ex. 20 15 13 11 8 7 6
There are 7 scores and the median is 11
2. When N is even, the median is the average of the two middle score.
Ex. 21 18 15 14 11 8 8 7
There are 8 scores and the median is (14+11)/2 = 12.5
3. When several scores have the same value as the midscore.
Ex. 15 15 14 11 9 9 9 6 5
Median is 9
Ex. 1. Find the median of the following set of observations.
8 4 1 3 & 7
Sol. Array the set of observations and find the median
1 3 4 7 8
4 is the middle item
2. Compute for the median from the following set of data
12 9 6 10 7 & 14
Array the data and computer for the median
6 7 9 10 12 14
Median = (9 + 10 )/2 = 9.5
Finding the Median from the Grouped Data
Md = L + [N/2 – F2 / f2] C
Where: L = lower class boundary of the interval where the median lies
N = No. of scores or sum of frequency
F2 = cumulative frequency less than up to the class immediately preceding
the median class (F<)
f2 = frequency of the median class
C= class size
Steps:
1. Prepare 3 columns (Class intervals, class frequency and cumulative frequency less than)
2. Determine the Median class. The median class is that class interval where n/2 lies.
3. Substitute the data to the formula.
Ex. Find the median of the frequency distribution
Weekly wages,No. of Workers(f) ,F< (cumulative frequency less than)
(in peso)
870 – 899, 4, 4
900 – 929 ,6, 10
930 – 959 ,10, 20 – F<
960 – 989, 13, 33 Median class
990 – 1019 ,8 , 41
1020 – 1049, 7, 48
1050 – 1079, 2, 50
N=50
To determine the median class:
Solve for N/2 = 50/2 = 25th
25th items fall in 960 – 989 class interval therefore it is the median class
Md = L + [(N/2 – F2)/f2] C
= 959.5 + [(50/2 – 20) /13] 30
Md = Php 971.04
When to use the median:
1. When the exact midpoint f the distribution is wanted the, 50% point.
2. When there are extreme scores which would markedly affect the mean. Extreme scores do not disturb the median.
3. When it is desired that certain scores should influence the central tendency, but all that is known about them is that they are above or below the median.
- determine of whether the cases fall within the lower halve or the upper halve of a distribution (appropriate locator of central tendency).
Finding the median from ungrouped data:
1. When N is odd, the median is the middle score.
Ex. 20 15 13 11 8 7 6
There are 7 scores and the median is 11
2. When N is even, the median is the average of the two middle score.
Ex. 21 18 15 14 11 8 8 7
There are 8 scores and the median is (14+11)/2 = 12.5
3. When several scores have the same value as the midscore.
Ex. 15 15 14 11 9 9 9 6 5
Median is 9
Ex. 1. Find the median of the following set of observations.
8 4 1 3 & 7
Sol. Array the set of observations and find the median
1 3 4 7 8
4 is the middle item
2. Compute for the median from the following set of data
12 9 6 10 7 & 14
Array the data and computer for the median
6 7 9 10 12 14
Median = (9 + 10 )/2 = 9.5
Finding the Median from the Grouped Data
Md = L + [N/2 – F2 / f2] C
Where: L = lower class boundary of the interval where the median lies
N = No. of scores or sum of frequency
F2 = cumulative frequency less than up to the class immediately preceding
the median class (F<)
f2 = frequency of the median class
C= class size
Steps:
1. Prepare 3 columns (Class intervals, class frequency and cumulative frequency less than)
2. Determine the Median class. The median class is that class interval where n/2 lies.
3. Substitute the data to the formula.
Ex. Find the median of the frequency distribution
Weekly wages,No. of Workers(f) ,F< (cumulative frequency less than)
(in peso)
870 – 899, 4, 4
900 – 929 ,6, 10
930 – 959 ,10, 20 – F<
960 – 989, 13, 33 Median class
990 – 1019 ,8 , 41
1020 – 1049, 7, 48
1050 – 1079, 2, 50
N=50
To determine the median class:
Solve for N/2 = 50/2 = 25th
25th items fall in 960 – 989 class interval therefore it is the median class
Md = L + [(N/2 – F2)/f2] C
= 959.5 + [(50/2 – 20) /13] 30
Md = Php 971.04
Thursday, May 15, 2008
Measures of Central Tendency - Mean
Central Tendency is the point about which the scores tend to cluster, a sort of average in the series. It is the center of concentration of scores in any set of data. It is a single number which represents the general level of performance of a group.
Three (3) measures of Central Tendency
a.) Mean – The mean on arithmetic mean, or arithmetic average is defined as the sum of the values in the data group divided by the no. of values.
When to use the mean
1. When the scores are distributed symmetrically around a central point.
2. When the measure of central tendency having the greatest stability is wanted.
3. When other statistics like standard deviation, coefficient of correlation, etc. are to be computer later, since these statistics are based upon the mean.
Finding the Mean from Ungrouped Data
Where: x = score or measure
X = ∑X /N N = No. of scores or measures
∑ = summation of
Example:
1.) Last year the five sales counselors of Pacific Plans Inc. sold the following number f educators plans; 24,16,35,13,25. Find the mean.
Solution:
X = (24+16+35+13+25)/5= 22.6
Finding the mean from Grouped Data:
Long method:
X = ∑f M / N
Where: f = class frequency
M = class midpoint
N = sum of the frequencies
By the “ Assumed mean” or short method :
X=AM+(∑f X / N)c
Where: AM = assumed mean
c = class size
x = deviation
Example: Scores of 50 students on a college algebra test
Class internal, Midpoint, fx, Class frequency f, x, fM
scores
45-47 ,46 ,18 ,3 ,6, 138
42-44 ,43, 20, 4 ,5 ,172
39-41 ,40, 16, 4, 4, 160
36-38 ,37 ,12, 4, 3 ,148
33-35 ,34, 4, 2 ,2 ,68
30-32 ,31, 3 ,3 ,1 ,93
27-29 ,28 ,-0 ,13, 0,364
24-26 ,25 ,-8, 8 ,-1, 200
21-23 ,22, -6, 3, -2, 66
18-20, 19, -9, 3, -3,57
15-17, 16, 0, 0 ,-4 ,0
12-14 ,13, -10, 2, -5, 26
9-11 ,10, -6, 1, -6 ,10
34 50 1502
Long method :
X = ∑f M/ N = 1502/50=30.04
Assumed mean or short method : Steps :
1. Prepare 4 column ( class interval,f,x,fx ).
2. Select the interval to contain the assumed mean (AM). For the assumed mean, one may take the midpoint of the interval near the center of the distribution, or the midpoint of the interval with the highest frequency.
3. Determine the x column starting with 0, number each class interval positive up to the highest class interval; negative up to the lowest class interval.
4. Multiply f by x to determine the fx column.
5. Find the algebraic sum of the positive fx’s and the negative fx’s to get ∑fx.
Short method :
X=AM+(∑f X / N)c=28+(34/50)3=30.04
Weighted Arithmetic mean or Combined mean
Where: w = weight of x
∑wx = sum of the weight of
∑w = sum of the weight of x
Example: The same test was administered to fourth year high school students in 3 schools. Each school had computed its own mean using internal width of 3 as shown below.
School A
Class Interval
Scores f, x, fx
39-41, 1, 4 ,4
36-38,2 ,3 ,6
33-35, 4 ,2, 8
30-32, 4, 1 ,4
27-29 2 0 0
24-26, 3, -1, -3
21-23 ,4 ,-2, -8
18-20,2,-3,-6
22 5
School B
42-44, 1, 6, 6
39-41, 0, 5,0
36-38, 2, 4 ,8
33-35 ,5 ,3, 15
30-32 ,6, 2, 12
27-29, 7, 1, 7
24-26 3 0 0
21-23, 4, -1, -4
18-20, 2, -2, -4
15-17 ,2, -3 ,-6
12-14 ,1, -4, -4
9-11, 2, -5, -10
35 20
School C
39-41, 1, 3, 3
36-38, 2 ,2 ,4
33-35 ,10, 1, 10
30-32, 6, 0 ,0
27-29 ,7, -1 ,-7
24-26, 2, -2, -4
21-23, 1, -3, -3
18-20, 0 ,-4, 0
15-17 ,1, -5, -5
30 -2
To find the weighted mean of the 3 schools, follow the procedures below:
1. Find the highest and the lowest scores of the schools.
2. Prepare the step intervals column for the combined distribution.
3. Write the frequencies for each steps interval for the three schools.
4. Find the total frequency for each steps interval for the total combined distribution.
5. Compute the mean from this distribution.
Class interval
Scores , sch.A (f) ,sch.B (f), sch.c(f) ,total f, x, fx
42-44, 0, 1, 0, 1, 5, 5
39-41 ,1, 0 , 1, 2 ,4,8
36-38 2,2,2,6,3,18
33-35 ,4, 5 ,10 ,19, 2,38
30-32 ,4, 6 ,6, 16, 1 ,16
27-29, 2, 7 ,7, 16, 0 ,0
24-26 ,3, 3, 2, 8, -1, -8
21-23, 4, 4, 1, 9, -2 ,-18
18-20, 2, 2, 0 ,4 ,-3 ,-12
15-17, 0, 2 ,1, 3 ,-4, -12
12-14 ,0 ,1, 0 ,1 ,-5, -5
9-11 ,0, 2, 0, 2 ,-6, -2
22, 35, 30, ,18
X=AM+(∑f X / N)c=28+(18/87)3=28.6
WX = ∑wx / ∑w =22(28.69)+35(26.71)+30(30.8)/87=28.62
Saturday, May 3, 2008
Graphical Method of Presenting Data and Frequency
1. Histogram. Class boundaries (x) vs. class frequency (y)
2. Frequency Polygon. Class Mark (x) vs. class frequency (y)
3. Less than ogive, upper class limit (x) vs. less than cumulative frequency (y)
4. Greater than ogive. Lower class limit (x) vs. greater than cumulative frequency (y)
2. Frequency Polygon. Class Mark (x) vs. class frequency (y)
3. Less than ogive, upper class limit (x) vs. less than cumulative frequency (y)
4. Greater than ogive. Lower class limit (x) vs. greater than cumulative frequency (y)
Friday, May 2, 2008
Other Definition of Terms
Array – This is the arrangement of data from the highest to lowest or from lowest to highest.
Range, R - is the difference between the highest and the lowest number.
Number of class- it depends on the size and nature of or class interval distribution. The no. of classes is determined into which the range will be divided. Usually, an effective no. of classes is somewhere between 4 and 20.
No. of classes = range / class size or class width +1
Note:
a.) If series contains less than 50 cases, 10 cases or less are just enough
b.) If series contains 50 to 100 cases, 10 to 15 classes are recommended
c.) If more than 100 cases, 15 or more classes are good
Class Limit – the end number of a class. It is the highest and the lowest values that can go into each class.
Class Size – the width of each class interval
Class Boundaries – are the “true” class limits defined by lower and upper boundaries. The lower boundaries can be determined by getting the average of the upper limit of a class and the lower limit of the next class. They can also be obtained by simply adding of a unit (0.5) to the upper limit and subtracting the same to the lower limit of each class.
Class Mark, M – also known as class Midpoint. It is the average of the lower and upper limits or boundaries of each class.
Class Interval – The range of values used in defining a class. It is simply the length of a class. It is the difference or distance between the upper and lower class boundaries of each class and is affected by the nature of the data and by the number of classes. It is a good practice to set up uniform class interval whenever possible for easier computation and interpretation.
Range, R - is the difference between the highest and the lowest number.
Number of class- it depends on the size and nature of or class interval distribution. The no. of classes is determined into which the range will be divided. Usually, an effective no. of classes is somewhere between 4 and 20.
No. of classes = range / class size or class width +1
Note:
a.) If series contains less than 50 cases, 10 cases or less are just enough
b.) If series contains 50 to 100 cases, 10 to 15 classes are recommended
c.) If more than 100 cases, 15 or more classes are good
Class Limit – the end number of a class. It is the highest and the lowest values that can go into each class.
Class Size – the width of each class interval
Class Boundaries – are the “true” class limits defined by lower and upper boundaries. The lower boundaries can be determined by getting the average of the upper limit of a class and the lower limit of the next class. They can also be obtained by simply adding of a unit (0.5) to the upper limit and subtracting the same to the lower limit of each class.
Class Mark, M – also known as class Midpoint. It is the average of the lower and upper limits or boundaries of each class.
Class Interval – The range of values used in defining a class. It is simply the length of a class. It is the difference or distance between the upper and lower class boundaries of each class and is affected by the nature of the data and by the number of classes. It is a good practice to set up uniform class interval whenever possible for easier computation and interpretation.
Monday, April 28, 2008
What is Frequency Distribution?
- A common, very helpful way to summarize data collections method shows the frequency (no. of occurrences) in each of several categories.
- Frequency Distribution can be summarize large volume of data values so decision makes can extract useful information directly from the collection.
- Frequency Distribution is a tabular arrangement of data showing its classification or grouping according to magnitude or size.
- Large masses of data presented without any arrangement or classification given very little information. In order for these data to give useful information they should be summarized or organized into a reduced form more appropriate for an effective analysis. One way of doing this is to summarize the data and compress them into a frequency distribution.
Classification of Variables
A. According to continuity of values
1. Continuous variables. These are variables that can take the form of decimals.
Example. Weight, length, height, school achievement.
2. Discrete or discontinuous variable. These are variables that can’t take the form of
decimals.
Example: number of students, number of houses, size of a family, etc.
B. According to scale and measurements
1. Nominal variable. This property allows one to make statements of similarities or
differences.
Example: sex- member of population may be classified as male or female,
socio-economic status – the member of the group may be classified
as those belonging to high, average or low socio-economic status
2. Ordinal variable. This variable refers to a property whereby members of a group are
ranked.
Example: one can judge and rank the contestants in a beauty contest.
3. Internal variable. This property allows one to make statements of equality of intervals.
Example: height, weight, temperature, test scores, etc.
4. Ratio variable. This property permits making statements of quality of ratios.
Example: If Cora is 48 yrs. old and Philline is 22 years old. Their ages can be expressed in the ratio of 48:22 or 24:11 (twenty-four is to eleven)
C. According to Functional Relationship
1. Independent variable. This is sometimes termed as predictor variable.
2. Dependent variable. This is sometimes called criterion variable.
Example: Academic achievement is dependent on I.Q . I.Q. is independent variable and academic achievement is the dependent variable
Example. Weight, length, height, school achievement.
2. Discrete or discontinuous variable. These are variables that can’t take the form of
decimals.
Example: number of students, number of houses, size of a family, etc.
B. According to scale and measurements
1. Nominal variable. This property allows one to make statements of similarities or
differences.
Example: sex- member of population may be classified as male or female,
socio-economic status – the member of the group may be classified
as those belonging to high, average or low socio-economic status
2. Ordinal variable. This variable refers to a property whereby members of a group are
ranked.
Example: one can judge and rank the contestants in a beauty contest.
3. Internal variable. This property allows one to make statements of equality of intervals.
Example: height, weight, temperature, test scores, etc.
4. Ratio variable. This property permits making statements of quality of ratios.
Example: If Cora is 48 yrs. old and Philline is 22 years old. Their ages can be expressed in the ratio of 48:22 or 24:11 (twenty-four is to eleven)
C. According to Functional Relationship
1. Independent variable. This is sometimes termed as predictor variable.
2. Dependent variable. This is sometimes called criterion variable.
Example: Academic achievement is dependent on I.Q . I.Q. is independent variable and academic achievement is the dependent variable
Methods of Collecting Data
a. Direct or Interview Method – This is a personal communication with the individual you want to interview.
b. Indirect or Questionnaires Method – This is done by sending questionnaires to the person from whom like to get the information.
c. Registration – Utilizing existing records
Example: records of births, marriages and deaths at the National Census of Statistics Office
(NCSO)
d. Observation – This can be done directly or indirectly.
e. Experiment – This is done by making or conducting scientific inquiry.
Sunday, April 27, 2008
Definition of Terms
Data
It is a facts or figures from which conclusions may be drawn. The statistical facts, historical facts, principles, opinions and item of various sources like scores, age, I.Q., Income, etc.
Data Collection and Presentation The data collected must be valid, reliable, relevant, and consistent with other information to the problem at hand. Data collected may be classified as primary, secondary, internal or external.
Primary Data
Refer to the data obtained directly from an original source by means of actual observations or by conducting interview. The direct source could be an individual or family groups, business entities or private and government agencies.
Secondary Data
Refer to data or information that come from existing records (published and/or unpublished) in usable form such as surveys, census, business journals and magazines, newspapers, commercial publications and other such as theses and dissertation and research papers, etc.
Internal Data
Data taken from the company’s own records of operations such as sales records, production records, personal records, etc.
External Data
Data that come from outside sources and not from the company’s own record.
Variable
It is a characteristic or phenomena which may take on different values. Example: weight, I.Q., and sex, age, marital status, eye color, etc.
Quantitative variable
If the outcomes are expressed numerically. Example: height, weight, age and numerical values.
Qualitative variable
It the outcomes refer to non-numerical qualities or attributes. Ex. Sex, marital status, eye color.
It is a facts or figures from which conclusions may be drawn. The statistical facts, historical facts, principles, opinions and item of various sources like scores, age, I.Q., Income, etc.
Data Collection and Presentation The data collected must be valid, reliable, relevant, and consistent with other information to the problem at hand. Data collected may be classified as primary, secondary, internal or external.
Primary Data
Refer to the data obtained directly from an original source by means of actual observations or by conducting interview. The direct source could be an individual or family groups, business entities or private and government agencies.
Secondary Data
Refer to data or information that come from existing records (published and/or unpublished) in usable form such as surveys, census, business journals and magazines, newspapers, commercial publications and other such as theses and dissertation and research papers, etc.
Internal Data
Data taken from the company’s own records of operations such as sales records, production records, personal records, etc.
External Data
Data that come from outside sources and not from the company’s own record.
Variable
It is a characteristic or phenomena which may take on different values. Example: weight, I.Q., and sex, age, marital status, eye color, etc.
Quantitative variable
If the outcomes are expressed numerically. Example: height, weight, age and numerical values.
Qualitative variable
It the outcomes refer to non-numerical qualities or attributes. Ex. Sex, marital status, eye color.
Division of Statistics
Statistics may be divided into:
Examples: the measure of location, measures of variability, skewness and kurtosis
1. Descriptive Statistics – which is concerned with the collection, classification, and presentation of data designed to summarize and describe the group characteristics of the data.
Examples: the measure of location, measures of variability, skewness and kurtosis
2. Inferential Statistics – refers to the drawing of conclusions or judgment about a population based or representative sample systematically taken from the same population. It’s aim is to give concise information about large group of data without dealing with each and every element of these groups. So that, if the sample taken is small, certain assumptions and inferences are made based on limited information and if the sample drawn is large, it may be treated as equal to that of the whole observation.
Populations and Sample
Population. It is the totality of all the actual or concerable objects of a certain class under consideration. It is a complete set of individuals, objects or measurements having some common observable characteristics.
Sample. It is a finite number of objects selected from the population.
Importance of Statistics
Statistics or statistical method is playing an important role in nearly all fields of human endeavor. The influence of statistics has spread out in almost all fields such as education, agriculture, business, psychology, economics, physics, government, chemistry, sociology, and other branches of science and engineering.
We are aware of the activities in our school. Our teacher give us the grades which are the result of our performance based in recitation, quizzes, tests, homework and other classroom activities. Evaluation of student’s performance in the classroom makes use of statistical method. The school administrator evaluates teachers’ performance, keeps record of no. of students enrolled, gathers information about the students and collect data pertinent to efficient operation, These data are then analyzed and interpreted such activities make use of statistics.
In the sciences, statistical methods are used in the formulation of laws, principles, and established facts. In companies, results of statistics serve as bases for the formulation of policies for a more efficient operation of the business.
Some of the most important subject areas which make use of statistical theory and techniques are as follows:
Biology – Research and experimentation in life processes of plants and animals to promote growth or prolong life
Education – Teaching – Learning processes, measurement and evaluation, educational studies, enrollment, management and finance.
Education – Teaching – Learning processes, measurement and evaluation, educational studies, enrollment, management and finance.
Engineering – Design and Test of performance and quality control
Sports – Points made out of so many attempts from the field or foul from the line such as in basketball, football, etc.
What is Statistics?
Statistics is not really a new subject. It has been existence since the earliest man on earth.
The earliest tribes have been using statistics in keeping records of the:
Statistics is a branch of mathematics which deals with the collection, organization, analysis and interpretation of numerical data which maybe used for prediction and verification of relationships among variables.
The earliest tribes have been using statistics in keeping records of the:
- No. of individuals in their tribes
- No. of animals they have
- No. of enemies they killed and other similar data.
The early astronomers, through careful observations and accurate recordings of data were able to predict the changing courses of the stars and other heavenly bodies.
Statistics is a branch of mathematics which deals with the collection, organization, analysis and interpretation of numerical data which maybe used for prediction and verification of relationships among variables.
Saturday, April 26, 2008
Course Outline
- Introduction what statistics is?, Some term and their definition used in statistics
- Methods of Collection of data and Kinds of sampling
- Kinds of data and their examples
- Classification of variable
- Frequency Distribution, and construction
- Graphical presentation for continuous data and Graphical Presentation for discontinuous data
- Measures of Central Tendency(Mean, Median, Mode)
- Quartiles, Percentiles, and Deciles
- Dispersion
- Standard score and normal curve
- Linear regression and coefficient of correlation
- Hypothesis testing and Measures of difference
Course Objectives
- Know the importance of collected and organized data to statistical analysis.
- Give a clear understanding to some terms to be used that will facilitate its computation in statistics.
- Learn and present a number of common statistical method for summarizing large data set.
- Give an insight how to find the appropriate measure of dispersion and how to examine several quantitative methods for describing the shape of distribution and how to interpret these statistics.
- Determine the importance of the normal distribution in the analysis and evaluation of every aspect of experimental data.
- Determine the difference among correlation, coefficient of correlation and regression.
- Know the role of probability to inferential statistics.
Course Description
The word statistics refers in common usage to numerical data but it has an additional meaning that is more specialized. Statistics also refer to the methodology for the collection, presentation and analysis of data and for the use of such data.
The collection and summarization of data are important first steps in the process of using data for analysis and decision making. The statistical methodology for analyzing data and making decisions is called statiscal inference. The logical foundation of statistical inference is the mathematical theory of probability. We define probability as the likelihood that a given event will occur relative to all other events that can occur. We usually define probabilities based on the number or frequency with which events can occur.
The collection and summarization of data are important first steps in the process of using data for analysis and decision making. The statistical methodology for analyzing data and making decisions is called statiscal inference. The logical foundation of statistical inference is the mathematical theory of probability. We define probability as the likelihood that a given event will occur relative to all other events that can occur. We usually define probabilities based on the number or frequency with which events can occur.
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