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

SAMPLING FOR STATISTICS GCE O LEVEL/IGCSE

SAMPLE SURVEY

The term survey has been defined as a means of collecting information to meet a definite need.

When a survey is carried out by a sampling method, it is called a sample survey. The main steps in a sample survey are to:

Clearly state the objectives of the survey;

1. Define the population we wish to study as clearly as possible;
2. Construct the sampling frame by clearly defining the sampling units;
3. Choose an appropriate sample design and proper sample size;
4. Organize a reliable field work to achieve the objectives of the survey;
5. Summarize and analyses the data.

SAMPLING BIAS

The word bias means a systematic component of error which deprives a survey result of representativeness. Bias is different from a random error in the sense that the random errors balance out in the long run while bias is cumulative and does not become less as the sample size increases. Bias may rises due to:#

1. Negligence and carelessness during sampling procedure.
2. Faulty planning of sampling.
3. Wrong selection of sample units.
4. Incomplete investigation and survey.
5. Framing of wrong questionnaire.

RANDOM SAMPLE

A sample is called a random sample if the probability of selection for each unit in the popub is known prior to sample selection.

SIMPLE RANDOM SAMPLING

A sample is defined to be a simple random sample (SRS) if it is selected in such a manner that (i) each unit in the population has an equal chance of being included in the sample. And (ii) each possible sample of the same size has an equal chance of being the sample selected. For example, in a class of 20 students, every student has an equal chance of getting A-grade exam.

SELECTION OF SIMPLE RANDOM SAMPLES (SRS)

A simple random sample can be selected by the following methods.

For example, in a class of 30 students, if we want to choose only one student for any particular sports. We place the slip of the name of each student in the bowl and then draw one from it. Therefore it is random.

STRATIFIED RANDOM SAMPLE

A sample is defined to be a stratified random sample if it is selected from a population which has been divided into a number of non-overlapping groups or sub-populations, called strata (the plural of stratum), such that part of the sample is drawn at random from each stratum.

MARKS	NUMBER OF STUDENTS
Below 50	70
50-75	80
75-100	100
TOTAL	250

A survey is to be taken to find methods to improve the local bus services. Show how to select a sample of 100 students from this population for the survey.

STRATA	MARKS	SAMPLE SIZE FROM EACH STRATUM
1	Below 50	70 x 100/250=28
2	50-75	80 x 100/250=32
3	75-100	100 x 100/250=40
	TOTAL	100

SYSTEMATIC RANDOM SAMPLE

A sample is defined to be a systematic sample if it is obtained by choosing one unit at random from the first k units and thereafter selecting every k^th unit in the population, serially numbered from 1 to N. The letter k is called the sampling interval.

Example: a particular type of bulb is made on a production line. Show how to select a systematic sample of 100 bulbs from 2000 bulbs produced on this production line on a certain day.

Solution: here we find the interval is of length 20, i.e. (2000/100). We begin buy selecting a bulb at random from the first 20 bulbs produced. If the 5^th bulb is chosen, then the other bulbs are selected at regular intervals, i.e. the 25^th, 45^th, 65^thi and so on until we have 100 bulbs.

QUOTA SAMPLING

A quota sample is a type of judgment sample. It is a sample, usually of human being, in which the information is collected purposively from the segments of a population (the quotas), e.g. the quota of men and women; urban and rural; upper, middle and lower income groups; etc.

For example, the class teacher may select the `well behaved' or `ill behaved^' from a particular class by their personal judgment.