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PROBABILITY FOR STATISTICS GCE O LEVELS/IGCSE

INTRODUCTION

The word probability has two basic meanings: (i) a quantitative measure of uncertainty and (ii) a measure of belief in a particular statement or problem. Probability and statistics are fundamentally interrelated

EXPERIMENT

A Process which gives some results.

RANDOM EXPERIMENT

An experiment which produced different results is called random experiment. The result obtained from an experiment or is called an outcome. Drawing of a card from a well shuffled deck of 52 playing cards are example of random experiment. A random experiment has three properties.

1. The experiment can be repeated any number of times.
2. The experiment always has two or more possible outcomes.
3. The out comes of each repetition is unpredictable.

SAMPLE SPACE

The set of all possible outcomes from a random experiment is called sample space.

EVENT

It is an individual outcome or any number of outcomes of a random experiment.

Simple event: An event that contains exactly one sample point is called simple event. Example in tossing coin head is an event and tail is also an event.

MUTUALLY EXCLUSIVE EVENTS

In some situations two events cannot occur at the same time. These events are called mutually exclusive events. In this case:

P(A ∩ B) = 0

P(A or B) = P(A) + P(B)

EXHAUSTIVE EVENT

Events are said to be exhaustive when the union of mutually exclusive events is the entire sample S.

INDEPENDENT EVENTS

When the probabilities of certain events occurring are quite unconnected to one another, these events are said to be independent events.

P(A ∩ B) = P(A) x P(B)

P(A and B) = P(A) x P(B)

Author: Saif Kheraj

TIME SERIES SCATTER DIAGRAM (SEMI AVERAGE) FOR STATISTICS GCE O LEVELS/IGCSE

INTRODUCTION

A time series consists of numerical data collected. observed or recorded at more or less regular intervals of time each hour, day, month, quarter or year. Examples of time series are the hourly temperature recorded, annual rain fall, etc.

SCATTER DIAGRAM

Scatter diagram is a graphic picture of the sample data. In scatter diagram different points are plotted. Due to fluctuations which includes seasonal, cyclical etc line is not drawn straight so we have to draw line of best fit.

TREND

A trend is a long term movement that persist for many years and indicates the general directions of the change of observed values.

GRAPH (LONG TERM CHANGE)

Source of image: https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUwz937f6zWacwfDTzc1BMT3Od-PRIl8yaXernjsNwGjC0WeGAdI7RfYqOJ6NJIiuNBdBxVcxrQsbDdhOHim02UStTf_Wz9IjweG60EpwhXttWmQge6gOkvnLbE3npB_e6aPr1FHX_hByb/s400/long-term+trends.png

SEASONAL VARIATIONS

Variation which is caused by the change in seasons. Prices of different goods and services changes with the change in season. The main cause for seasonal variation are weather conditions, festival and customs.For example sale of ice cream are high in

the summer season, sale of sweaters are high in the winter season and low in the summer season. This represents seasonal fluctuation.

CYCLICAL VARIATIONS

These fluctuations occur around a long-term growth trend.There are two periods in the business cycle one is boom when economic activities and growth is high with increase demand for goods and services. This increases sales and business activity. Other period is recession when economic activity is down, prices decreases due to decrease demand for goods and services unemployment increases and economic activity slows down.

This shows economic cycle you can see fluctuation in the 2 periods.

ANALYZING THE TREND

Two Methods are used

1. Semi averages method
2. Moving averages method

THE METHOD OF SEMI AVERAGES

Divide the values in the series in the two equal parts. Find the averages of the values in each pa

rt and plot the average values against the midpoint of two parts. Then draw the straight line.

This straight line can be described by the mathematical equation

y=mx+c

m=gradient (RATE)

c=y intercept

x and y=constant

EXAMPLE

Year	Profit	Total	Averages
2005	10
2006	50	210	70
2007	150
2008	80
2009	120	300	100
2010	100

GRAPH

Finally we will draw the graph. First we will plot all the points given in the question, in this question year and profits were given so plot out the points you can see in blue and then join together. Next we will plot the two points (averages) we have calculated and join it together with a straight line (Not drawn in the below diagram). In this diagram it is represented by pink.

Ex: Now join these two pink points together making a straight line

Notes: If you have further query or need any additional resources please comment. Your suggestions will be highly appreciated.

INDEX NUMBERS FOR STATISTICS GCE O LEVELS/IGCSE

INTRODUCTION

Index number is a statistical measure of average change in a variable or a group of variables with respect to time or space. The variable may be the cost of commodity or goods. 2 years are compared to see the changes in price. Index number generally computed on annual basis.

Definition:

Index number is a statistical measure that is used to show changes in price, quantify or value of an item or group of related items with respect to time, place or other characteristics.

In order to calculate an index number, a base period needs to be identified. Values in the current period are than compared to the base period.

The index numbers for the base period is 100. The index number for the current period is

then compared to 100.

SIMPLE PRICE INDEX

Index number is called a simple index when it is computed for a single variable. Index number on index number of gold prices, etc

Simple Price Index= P1/P2 x 100

P1=price of item in a given year

P2=price of item in a base year

PRICE RELATIVE

The base year is the particular year with which the prices in other years of that commodity can be compared with. A relative price is the price of a commodity such as a good or service in terms of another; i.e., the ratio of two prices. A relative price may be expressed in terms of a ratio between any two prices or the ratio between the price of one particular good and a weighted average of all other goods available in the market. Source= http://en.wikipedia.org/wiki/Relative_price

SIMPLE AGGREGATE INDEX

It is useful when finding index for group of item. It is one that indicates the percentage changes in the aggregate price of a number of Commodities, at different periods.

Simple aggregate index = ∑p₁/∑p₀ x 100

∑p₁=Total of prices of all commodities in given year

∑p₀ = Total of prices of all commodities in base year

Example

Cost of foods are shown below

	vegetable	Rice	Milk	Total
2010	$50	$20	$10	$80
2011	$55	$25	$8	$88

Using 2010 as a base year, find the simple aggregate index for the total cost for 2011.

Simple aggregate index =∑p₁/∑p₀ x 100

Simple aggregate index =88/80 x 100

Simple aggregate index =110

This shows that price has rise by 10 percent in the 2011 compared to 2010.

WEIGHTED AGGREGATED PRICE INDEX

An index is called a weighted aggregative index when it is constructed for an aggregate of items (prices) that have been weighted in some way so as to reflect their importance.

Weighted aggregate index number = ∑IW/∑W

∑IW= Price of item in given year.

∑W= Price of item in base year.

W= Weights

I=Price Relative

How to calculate Weights

By taking ratios of the

1. Expenditure on different items.

2. Quantities used of different items

Example

	2010	2011	Weights for 2010	Price relative(I)	IW
X	20	35	8	35/20 x 100=175	1400
Y	30	40	4	40/30 x 100=133.33	533.32
Z	60	64	3	64/60 x 100=106.67	320.01
A	90	94	6	94/90 x 100=104.44	626.64
B	20	40	10	40/20 x 100=200	2000
TOTAL			31		4879.97

Weighted aggregate index number = ∑IW/∑W

Weighted aggregate index number =4879.97/31

Weighted aggregate index number =157.42

USES OF INDEX NUMBER

1. The price index numbers are used to measure changes in a particular group of prices and help us in comparing the movement in prices of one commodity with other.
2. A common use of index number is to deflect a future value so that it can be compared with the base period.
3. They are also used to forecast business condition of a country.
4. It is a statistical device used by the government to revise wages, salaries, pensions, social welfare schemes and design future planning.

RATES ( DEATH RATE) FOR STATISTICS GCE O LEVELS/IGCSE

RATIO

A ratio expresses the relation of a given kind of event to the occurrence of other events

Ratio= x/y or x:y

The important types of ratios are the death ratios, birth ratios, etc.

RATE

In mathematics, a rate is a ratio between two measurements, often with different. Source Wikipedia (Rate).

MORE ON RATES VISIT http://en.wikipedia.org/wiki/Rate_(mathematics)

CRUDE DEATH RATE

The crude death rate defined as a ratio of total deaths of some specific year to the total population in the same year, multiplied by 1000. It computed as follows;

CRUDE DEATH RATE= Deaths/Population x1000

AGE SPECIFIC DEATH RATE

When death rate is computed for some specific class or specific age group of a population, it is called age specific death rate. The kind of specificity must be stated.

STANDARD POPULATION

The standard population is that population which is selected to be the basis of comparison.

STANDARDIZED DEATH RATE

The crude death rates of two localities cannot be compared because death rates differ with a age, sex, climate, occupation, etc. To eliminate such effects, we compute what are standardized death rates.

SDR = Expected deaths in standard population/Total standard population x 1000