1 Faculty of Social Sciences Induction Block: Maths & Statistics Lecture 2 Algebra and Notation Dr Gwilym Pryce

2 Plan n 1. Integers, Fractions, Percentages and decimals n 2. Adding variables n 3. Multiplying variables n 4. Multiplying a variable by itself n 5. Exponents and Logs n 6. Subscripts n 7. Summation sign

3 1. Integers, Fractions, Percentages and decimals n An Integer is a whole number –e.g. 2, or 7, or 503 n 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

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

5 n A decimal fraction has as its denominator a number which is a power of 10 (e.g. 100 which is 10 squared = 10 2 ) –e.g. 3/10 –e.g. 4/100 –e.g. 5/1000

6 n 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 10 1 I.e. denominator = 10 1 = 10 e.g. 3/10 = 0.3

7 –two places to the right of the decimal point means dividing by 10 2 I.e. denominator = 10 2 = 100 e.g. 4/100 = 0.04 –three places to the right of the decimal point means dividing by 10 3 I.e. denominator = 10 3 = 1000 e.g. 5/1000 = 0.005

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

9 2. Adding variables n x + y

10 3. Multiplying variables n xy

11 4. Multiplying a variable by itself nx2nx2

12 5. Exponents and Logs n Exponent: raising a constant or a variable to the power of a variable: –Constant raised to the power of a variable: e.g. 4 x e.g x = e x = exp[x] = x –Variable raised to the power of variable e.g. y x

13 6. Subscripts abbreviation for any six observations (numbers) is x 1, x 2, x 3, x 4, x 5, x 6 this can be abbreviated further as x i = x 1, …, x n where n = 6.

14 7. Summation –mean –standard deviation

15 E.g. Mean n sum of values divided by no. of values: e.g. mean of six numbers: 1,3, 8, 7, 5, 3 = ( ) / 6 = 4.5 n Algebraic abbreviation: abbreviation for sample mean is x-bar abbreviation for sum is capital sigma abbreviation for any six observations (numbers) is x 1, x 2, x 3, x 4, x 5, x 6 this can be abbreviated further as x i = x 1, …, x n where n = 6.

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17 n sample mean: n Population mean:

18 E.g. Variance n 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 “s 2 ”

19 n Q/ Why not simply take the average deviation? –I.e. why square the deviations first? n A/ sum of deviations from mean always = 0 –positive deviations cancel out negative deviations. n But if we square deviations first, all become positive.

20 E.g. Standard Deviation n 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. n This problem is overcome by taking the square root of the variance:

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22 Trimmed Mean: n 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.