Mathematical Content & Computation (AQA GCSE Psychology): Revision Note

Decimal & standard form

• Decimals are any numbers which include a decimal point, for example:

  • 6.31

  • 20.059

  • 468.27

• The digits before the decimal point are whole numbers; the digits after the decimal point are parts of that whole number, e.g.

  • 6.31: the 6 in this number refers to 6 units; 3 in this number refers to 3 tenths

  • 20.059: the 2 refers to two tens; the 5 refers to 5 hundredths, and the 9 refers to 9 thousandths

  • 468.27: the 4 refers to 4 hundreds; the 2 refers to 2 tenths

• Standard form is a way of dealing with very large (or very small) numbers without calculations becoming too cumbersome, e.g.

  • 10² = 100 (10 x 10)

  • 835,000,000,000 = 8.35 × 10¹¹ in standard form (835 is reduced to a number between 1 and 10, then '10 to the power of' expresses the size of the number)

• Small numbers can also be written in standard form

• In this case the index (the ‘to the power of’ number) must be negative, e.g.

  • 0.000000000000761 is written as 7.61 × 10^⁻¹³

Fractions & ratios

• Fractions enable researchers to see parts of the whole in terms of the data set they have collected, e.g.

  • 5 out of 25 participants scored above 100 in a concentration task = 5/25

  • 20 out of 50 participants stated that purple was their favourite colour = 20/50

• Fractions should be reduced to their simplest form by finding the highest common factor between the numerator and denominator and dividing both by that factor, e.g.:

  • 5/25 = 1/5 (5 is the highest common factor)

  • 20/50 = 2/5 (10 is the highest common factor)

• To change a fraction into a decimal, divide the numerator by the denominator, e.g.

  • 1/5 = 1 ÷ 5 = 0.2

  • 2/5 = 2 ÷ 5 = 0.4

• Ratios enable researchers to compare quantities as proportions of the whole data set, e.g.

  • 5 out of 25 participants scored above 100 in a concentration task = 5:25

  • 20 out of 500 participants stated that purple was their favourite colour = 20:50

  As with fractions, ratios should be reduced to their simplest form, e.g.

  • 5:25 = 1:5

  • 20:50 = 2:5

Percentages

• A percentage is a number or quantity calculated as a proportion out of 100, e.g. 65%, 3%, 18%

  Percentages can be expressed as a fraction or a decimal, e.g.:

  • 65% as a decimal is 0.65; as a fraction it is 13/20

  • 3% as a decimal is 0.03; as a fraction it is 3/100

  To calculate a percentage from a data set, multiply the numerator by 100 and divide by the denominator, e.g.:

  • 63 out of 70 participants chose A: 63 × 100 = 6300 ÷ 70 = 90%

  • 15 out of 82 participants scored below average: 15 × 100 = 1500 ÷ 82 = 18.29%

Significant figures & estimating results

• Significant figures is another way of dealing with very large (or very small) numbers

• A very large number can be rounded to the nearest round number, e.g.:

  • 596,321 rounded to one significant figure = 600,000

  • 341,602 rounded to one significant figure = 300,000

• For numbers with a decimal point, the digits after the decimal point are rounded up or down depending on the number of significant figures required, e.g.:

  • 0.00038967 to two significant figures = 0.00039

  • 0.0000578 to two significant figures = 0.000058

• A common mistake is confusing decimal places with significant figures:

  • Decimal places are counted from just after the decimal point

  • Significant figures are counted from the first non-zero digit, wherever it appears in the number

• Estimating results can be done by rounding numbers up or down before carrying out a calculation, e.g. 619 × 280 could be estimated as 600 × 300