Chapter 9: Apply statistics methods

9.1 Collection of raw data

Raw data are the observations and measurements exactly as they were first recorded, before they are sorted, summarised or analysed. In biology and laboratory work they include the number of colonies on a plate, the mass of a specimen on a balance, a reading from a pH meter, the number of insects in a trap and the length of a cell under a microscope. Statistics methods can give a sound answer only if the raw data were collected carefully, because no calculation can correct data that were measured badly or taken from an unrepresentative sample. Raw data are the starting point of every method in this chapter.

Sources of data

• Primary data: collected by the person using them, for example counts made on the technician's own plates or readings taken from the technician's own instrument. The collector knows how each value was obtained.
• Secondary data: collected earlier by someone else, such as published tables of measurements, an instrument's stored records or a colleague's results. Their method of collection has to be checked before they are used.

Methods of collecting data

• Counting: the number of individuals in a defined area, volume or time, for example colonies per plate, cells per field of view or organisms per quadrat. Counts give whole numbers.
• Measuring: a quantity read from an instrument, such as mass from a balance, length from a ruler or an eyepiece micrometer, volume from a pipette or burette, and temperature or pH from a meter. Measurements can take any value within the instrument's precision.
• Observation: recording what is seen, such as the colour of a colony or the presence or absence of a feature, in an agreed set of categories.
• Experiment: deliberately changing one factor, for example the temperature of an incubation, and recording the response while other factors are held constant.
• Records: using the readings logged by instruments or by earlier laboratory work, which counts as secondary data.

Sampling

It is rarely possible to measure every individual in a population, so a sample is measured. A sample must be chosen without bias: in a field plot, quadrats are placed at positions given by random number pairs rather than where the vegetation looks interesting, and in a laboratory the samples are picked by a random draw. A larger sample describes the population more reliably than a small one. Replicates, which are repeated measurements under the same conditions, show how much the values vary by chance.

Recording and quality of raw data

• Prepare a record sheet before starting, with a column for each item and a heading with the unit, for example length in millimetres.
• Record each value immediately in ink, and cross out a mistake with a single line so that it stays readable.
• Write each value to the precision of the instrument and keep the same number of decimal places throughout.
• Note the date, the sample identity, the instrument used and the person recording.
• Keep the original sheet. Copies or calculated values never replace the raw record.

9.1.1 Ungrouped data

Ungrouped data are raw data listed as individual values, with no classes. They suit small sets of values and counts that take only a few different whole numbers. The values are first sorted, and then a frequency table shows how many times each value occurs. The frequency of a value is the number of times it occurs.

Example: the number of aphids on each of 20 leaves in a field plot is recorded in the order of collection:

3  5  4  2  5  6  3  4  4  5  2  3  4  5  4  6  3  4  5  4

Making a frequency table from ungrouped data

1. List every different value in order from smallest to largest, here 2, 3, 4, 5 and 6.
2. Go through the data in order. For each value make one stroke beside it in a tally column; the fifth stroke is drawn across the four before it, so that the tally is read in fives.
3. Count the strokes beside each value and write the total in a frequency column.
4. Add the frequencies and check that the sum equals the number of observations, 20.

The table shows at a glance that 4 aphids per leaf occurs most often, while counts of 2 and 6 are least frequent. Ungrouped data keep every detail of the original values, which is why they are used when there are only a few different values.

9.1.2 Grouped data

Grouped data are raw data sorted into classes, which are intervals of values. Grouping is used when there are many observations or many different values, and especially for measurements that can take any value, such as lengths and masses, where listing every value would produce a table nearly as long as the raw data. Grouping makes the pattern easy to see, but it loses the exact values inside each class.

Example: the root length in millimetres of 30 seedlings, measured to the nearest millimetre:

23 35 41 28 52 38 44 31 47 26 39 33 58 42 36
29 45 50 34 40 27 37 46 32 55 43 30 48 36 41

The smallest value is 23 mm and the largest is 58 mm.

Terms used in grouped data

• Class interval: the range of values in one class, written 20 – 29 for values from 20 to 29 inclusive.
• Class limits: the smallest and largest values that can be written in a class, 20 and 29 for the first class.
• Class boundaries: the points midway between the upper limit of one class and the lower limit of the next. For measurements to the nearest millimetre, the first class runs from 19.5 to 29.5.
• Class width: the upper boundary minus the lower boundary, 29.5 − 19.5 = 10.
• Class midpoint: the value halfway between the class limits, (20 + 29) ÷ 2 = 24.5, used to stand for every value in the class.
• Frequency: the number of observations that fall in the class.

Steps for grouping raw data

1. Find the smallest and largest values and subtract to get the range: 58 − 23 = 35 mm.
2. Choose a number of classes, usually between 4 and 10, and a convenient equal width that covers the range, here a width of 10 mm giving 4 classes.
3. Write the classes so that they do not overlap and leave no gaps: 20 – 29, 30 – 39, 40 – 49 and 50 – 59.
4. Go through the data and tally each value in its class.
5. Count the tallies and write the frequencies.
6. Add the frequencies and check that the total equals the number of observations, 30.

The 30 – 39 mm class holds the most seedlings, 11 of the 30. The same table is the starting point for the charts and measures in the topics that follow.