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Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 2

Algebra and Notation

Dr Gwilym Pryce

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Plan 1. Integers, Fractions, Percentages and

decimals 2. Adding variables 3. Multiplying variables 4. Multiplying a variable by itself 5. Exponents and Logs 6. Subscripts 7. Summation sign

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1. Integers, Fractions, Percentages and decimals

An Integer is a whole number– e.g. 2, or 7, or 503

A fraction is the ratio of two numbers or variables:– I.e. it is one number or variable (called the

numerator) divided by another (called the denominator)

– e.g. 1/3– e.g. x/y

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A proper fraction is one were the numerator is less than the denominator– e.g. 1/3

An improper fraction is a fraction where the numerator is greater than the denominator and can be expressed as a mixed number:– e.g. 4/3 = 11/3

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A decimal fraction has as its denominator a number which is a power of 10 (e.g. 100 which is 10 squared = 102)– e.g. 3/10– e.g. 4/100– e.g. 5/1000

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Using the decimal point notation means that the denominator can be omitted for sake of brevity:– one place to the right of the decimal point

means dividing by 101

• I.e. denominator = 101 = 10• e.g. 3/10 = 0.3

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– two places to the right of the decimal point means dividing by 102

• I.e. denominator = 102 = 100• e.g. 4/100 = 0.04

– three places to the right of the decimal point means dividing by 103 • I.e. denominator = 103 = 1000• e.g. 5/1000 = 0.005

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Percentage is a way of representing a number as a fraction of 100:– e.g. 45 percent = 45% = 45/100 = 0.45– e.g. 125 percent = 125% = 125/100 = 1.25

Decimals can be written as percentages by multiplying by 100:– e.g. 0.3 = 30%– e.g. 0.04 = 4%– e.g. 0.005 = 0.5%

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2. Adding variables

x + y

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3. Multiplying variables

xy

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4. Multiplying a variable by itself

x2

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5. Exponents and Logs

Exponent: raising a constant or a variable to the power of a variable:– Constant raised to the power of a variable:

• e.g. 4x

• e.g. 2.71828x = ex = exp[x] = 2.71828x

– Variable raised to the power of variable e.g. yx

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6. Subscripts

• abbreviation for any six observations (numbers) is x1, x2, x3, x4, x5, x6

• this can be abbreviated further as xi = x1, …, xn where n = 6.

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7. Summation

– mean– standard deviation

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E.g. Mean sum of values divided by no. of values:

• e.g. mean of six numbers: 1,3, 8, 7, 5, 3= (1 + 3 + 8 + 7 + 5 + 3) / 6 = 4.5

Algebraic abbreviation:• abbreviation for sample mean is x-bar• abbreviation for sum is capital sigma• abbreviation for any six observations (numbers)

is x1, x2, x3, x4, x5, x6

• this can be abbreviated further as xi = x1, …, xn where n = 6.

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n

xx i

3,5,7,8,3,1 where,6

i

i xx

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3 5 7 8 3 1 average mean

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sample mean:

Population mean:n

xx i

N

X i

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E.g. Variance Based on the mean:

– sum of all squared deviations from the mean divided by the number of observations

– “average squared deviation from the average”

– denoted by “s2”

1

)(...)()( 222

212

n

xxxxxxs n

22 )(1

1xx

ns i

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Q/ Why not simply take the average deviation?– I.e. why square the deviations first?

A/ sum of deviations from mean always = 0 – positive deviations cancel out negative

deviations. But if we square deviations first, all

become positive.

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E.g. Standard Deviation Problem with the variance is that it’s

value is sensitive to the scale of the variable.– E.g. variance of incomes measured in £will

be much greater than the variance of incomes measured in £000.

This problem is overcome by taking the square root of the variance:

2)(1

1 Deviation Standard xx

ns i

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Observation Price per night of 3 star London hotelsx i x xxi ( xxi )2

1 £ 67.20 £ 71.01 -£ 3.81 £ 14.502 £ 70.49 £ 71.01 -£ 0.52 £ 0.273 £ 78.26 £ 71.01 £ 7.25 £ 52.594 £ 65.80 £ 71.01 -£ 5.21 £ 27.125 £ 70.56 £ 71.01 -£ 0.45 £ 0.206 £ 83.23 £ 71.01 £ 12.22 £ 149.387 £ 80.01 £ 71.01 £ 9.00 £ 81.048 £ 66.99 £ 71.01 -£ 4.02 £ 16.149 £ 73.15 £ 71.01 £ 2.14 £ 4.59

10 £ 54.39 £ 71.01 -£ 16.62 £ 276.16Sum: £ 0.00 £ 621.99

Sum / (n-1) 0 £ 69.11(Variance)

(sum / (n-1)) 8.31(StandardDeviation)

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Trimmed Mean: Percentiles:

– trimmed mean = mean of the observations between the third and the first quartiles.

– Outliers and extreme observations do not affect this alternative measure of the mean.