Ways of writing one result

1. A set of results can be reported as a fraction, a decimal, a percentage, a ratio or in standard form, and the form chosen is the one that makes the finding clearest.
2. One value can take several of those forms at once, so 12 correct out of 40 is 3/10, 0.3 and 30%.
3. Spread, precision, the shape of a distribution and estimation are all reported alongside the mean, median and mode.

Fractions, decimals and percentages

1. Fraction: the part written over the whole, so 12 correct out of 40 is 12/40.
   1. Divide top and bottom by the same number to simplify, so dividing 12/40 by 4 gives 3/10.
2. Decimal: divide the top of the fraction by the bottom, so 12 divided by 40 gives 0.3.
3. Percentage: a part out of 100, found by multiplying the decimal by 100, so 0.3 becomes 30%.
   1. To reverse that step, divide the percentage by 100, so 30% goes back to 0.3.
4. A percentage lets two groups of different sizes be compared, because 12 out of 40 and 30 out of 100 are both 30%.

Ratios

1. A ratio compares one part with another part rather than with the whole, written with a colon between the two amounts.
2. Simplify a ratio the way a fraction is simplified, by dividing both sides by a common factor.
3. Keep the amounts in the order the question names them, so passes to fails is 12:28 and not 28:12.

Standard form

1. Standard form writes an awkward number as a value between 1 and 10 multiplied by a power of ten.
2. Move the decimal point until one non-zero digit sits in front of it, then count the places it moved.
3. Moving the point left gives a positive power, and moving it right gives a negative power.
4. Psychology uses it for large samples and for very short times, both of which are clumsy written out in full.

Significant figures and decimal places

1. Significant figures are counted from the first non-zero digit, whichever side of the decimal point that digit falls on.
2. Decimal places are counted only from the point onwards, so they ignore every digit in front of it.
3. The two counts usually give different answers for the same number.
4. In 7.42 there are three significant figures and two decimal places.
5. In 0.0035 the first significant figure is the 3, giving two significant figures and four decimal places.
6. A result is reported to a precision the data can carry, because spare digits look exact without being exact.

Rounding

1. Find the place being rounded to, then look at the single digit immediately after it.
2. Round up when that digit is 5 or more, and leave the digit unchanged when it is less than 5.
3. Round once at the end of a calculation, because rounding part way through changes the final answer.

Choosing a level of precision

1. The precision of an answer matches the precision of the original measurements.
2. A count of people finishes as a whole number, because there is no such thing as part of a participant.
3. Reaction times timed to a tenth of a second stay at one decimal place, however many digits the calculator offers.

The range

1. The range measures spread, also called dispersion, so it answers a different question from an average.
2. It is calculated by subtracting the lowest value from the highest value.
3. The larger the range, the more widely the scores are spread.
4. Two classes can share one mean and still have very different ranges, which is why the two figures are reported side by side.
5. The range is reported in the same units as the data, so a spread of 9 words recalled is written as 9 words.

Outliers

1. An outlier is a value far away from the rest of the data.
2. Because the range uses only the two end values, a single outlier controls it.
3. In the set 3, 5, 5, 6, 8, 9 and 12 the range of 9 comes from the lone score of 12, while five of the seven scores sit within 4 words of each other.
4. Taking that outlier out leaves six scores running from 3 to 9, giving a range of 6 words instead of 9.
5. When two ranges are compared, an extreme score in either set can be the reason they differ.

Normal distribution

1. A normal distribution is a symmetrical, bell-shaped spread of data with one peak in the middle.
2. Most scores cluster near the centre and the numbers thin out steadily towards each extreme.
3. Its defining feature is that the mean, median and mode all take the same value and sit under the peak.
4. Human characteristics such as height, reaction time and IQ follow the pattern once the sample is large enough.
5. Recognising the shape shows which scores count as typical and how rare a score at either extreme is.

Recognising a skewed distribution

1. A skewed distribution has its peak towards one end rather than in the middle, so the curve is not symmetrical.
2. In a skewed set the mean, median and mode no longer share a value.
3. The mean shifts furthest towards the long tail because it uses every score, so a gap between the mean and the mode is a sign of skew.

Estimation

1. Estimating means rounding the data to easy numbers and working out a rough answer before the exact sum.
2. Round each value to a number that can be added or multiplied mentally, usually the nearest ten.
3. An expected value can be estimated from a proportion the same way, so a quarter of 200 people gives about 50.
4. The estimate and the exact answer are then held next to each other.
5. A close match between them indicates the exact calculation is sound.
6. A large gap points to a misplaced decimal point, a missing zero or a division done the wrong way round.
7. The exact calculation gives the result, and the estimate only checks it.

Turning this into marks

1. When a figure and its meaning are both asked for, keep them as two separate points, one giving the calculation and one saying what the figure shows about the data.
2. Name whether significant figures or decimal places are wanted, because 7.4166 gives 7.42 to two decimal places and 7.4 to two significant figures.
3. Show the working line, so 12 divided by 40 equals 0.3, which is 30%, rather than the percentage on its own.
4. The mark most often lost is quoting the range without its units.