Mathematical Content & Computation (AQA GCSE Psychology)

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 in this number refers to two tens; the 9 refers to 9 hundredths

  • 468.27 - the 4 in this number refers to 4 hundreds; the 2 in this number refers to 2 tenths

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

  • 10 to the power of 2 = 100, which is written as 10²  (i.e. it refers to 10 x 10)

  • 835,000,000,000 = 8.35 × 10¹¹ in standard form (835 must be reduced to a number between 1 and 10 and then 10 ‘to the power of’ is added to express the number)

• Small numbers can also be written in standard form, however, 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

  • 16 out of 100 participants stated that purple was their favourite colour = 16/100

• Fractions should be reduced to their simplest form which is done by finding the highest common factor between the top (the numerator) and bottom number (the denominator) and dividing them by the factor, e.g.

  • 5/25 = ⅕ (5 is the common factor; it divides equally into 5 and 25)

  • 16/100 = 4/25 (4 is the highest common factor as 16 does not divide equally into 100)

• If you want to change a factor into a decimal number you need to divide the numerator by the denominator e.g. for ⅕ it is 1 ÷ 5 = 0.02; for 4/25 it is 4 ÷ 25 = 0.16

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

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

  • 16 out of 100 participants stated that purple was their favourite colour = 16:100

• As with fractions, a ratio should be reduced to its simplest form, e.g.

  • 5:25 = 1:5

  • 16:100 = 4:25

Percentages

• Percentages refers to 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 the percentage from a data set the numerator is multiplied by 100 and then divided by the denominator, e.g.

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

  • 15 out of 82 participants scored below average = 15 x 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 up to the nearest round number (a number with that ends with a 0), e.g.

    • 596,321 would be rounded up to 600,000

    • 341,602 would be rounded down to 300,000

  • For numbers with a decimal point it is the digits after the decimal point that are rounded up or down, e.g.

    • 0.00038967 to two significant figures is 0.00039 

    • 0.0000578 to two significant figures is 0.000058

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

    • Decimal places are rounded from just after the decimal point

    • Significant figures are rounded from the first digit which is not a zero, wherever it may fall in the number

• Estimating results can be done by rounding up or down, depending on the numbers involved in the estimation, e.g.

  • 619 x 280 could be estimated as 600 (rounding down to the significant figures involved) multiplied by 300 (rounding up due to the significant figures involved)