Faculty of Social Sciences Induction Block: Maths

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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 expx 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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• sample mean
• Population mean

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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

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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 its 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

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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.