Chapter 11: Apply statistics methods

11.1 Collection of raw data

Statistics is the science of collecting, organising, summarising and interpreting numerical information. Raw data are the values exactly as they were first recorded, before anything has been sorted, grouped or averaged: the burette readings in a titration logbook, the absorbance values printed by a spectrophotometer, the number of samples received each day. Everything later in a statistical analysis depends on the quality of these first records. A wrong value recorded at the bench cannot be repaired by a correct calculation later.

Population and sample

The population is the whole set of items that a question is about, for example every bottle in a production batch of cleaning solution. A sample is the part of the population that is actually measured, for example 12 bottles chosen from the batch. Samples are used because measuring the whole population is usually impossible, too slow, or destroys the items. A sample describes the population well only if it is chosen fairly. Choosing items at random, or at fixed intervals along the production line, gives every item a chance of being tested, whereas always picking the nearest bottle can give a biased sample.

Types of data

Sources and methods of collection

• Direct measurement: values read from a balance, burette, pH meter or other instrument and written in a logbook or saved by the instrument.
• Observation: counting or noting events, such as bubbles per minute or the number of samples arriving.
• Records: values taken from existing laboratory registers, calibration logs and delivery notes.
• Questionnaires and interviews: answers collected from clients or staff, for example about turnaround time.

Good practice when recording raw data

1. Decide before starting what will be measured, in what units, and how many measurements are needed.
2. Prepare a data sheet with labelled columns for sample identity, date and time, value, units and analyst.
3. Record each value at the moment of measurement, to the full resolution of the instrument. A balance that shows 0.0001 g is read to four decimal places.
4. Never rub out or overwrite a value. A wrong entry is crossed out with one line and the correction written beside it, with a reason.
5. Note the conditions that could affect the result, such as the temperature, the instrument used and the date of its last calibration.
6. Keep the original record safe and work from a copy when sorting or calculating.

11.1.1 Ungrouped data

Ungrouped data are raw values listed individually, one value for each observation, with no classes. They are suitable for small sets of data, or for discrete data in which the same few values repeat. When the same value occurs many times, the data are organised in a frequency table: one row for each distinct value, showing how many times it occurs. A tally is used while counting, and the frequency is the final count.

Tally and frequency table

A tally mark is a single stroke for each observation. Every fifth stroke crosses the previous four, so that the marks can be read in fives. In this text each stroke is written I, and the crossing fifth stroke is written as a slash after four strokes (IIII/ is five). The steps for making a frequency table are:

1. List every distinct value that occurs, in increasing order, in the first column.
2. Go through the raw data once, in order, and put one tally stroke against the value for each observation.
3. Count the strokes in each row and write the frequency in the next column.
4. Add the frequencies. The total must equal the number of observations in the raw data.
5. Check the total against the original data before using the table.

Worked example

Example: The number of defective filter papers in each of 30 boxes of 10 was recorded as follows:

2, 0, 1, 1, 3, 2, 0, 4, 1, 2, 1, 0, 3, 2, 1, 1, 0, 2, 3, 1, 2, 4, 0, 1, 3, 2, 1, 0, 4, 2

The total of 30 equals the number of boxes. The most frequent value is 1, found in 9 boxes. The total number of defective papers is 0 × 6 + 1 × 9 + 2 × 8 + 3 × 4 + 4 × 3 = 0 + 9 + 16 + 12 + 12 = 49.

11.1.2 Grouped data

Grouped data are values that have been sorted into classes, which are intervals, with a frequency recorded for each class. Grouping is used when there are many different values, as with continuous measurements. A list of 40 chloride results with few repeated values would give 40 rows in a frequency table, and the pattern would be hard to see, but five or six classes show it clearly. The price of grouping is that the individual values are lost, so later calculations use the class mid-point instead of the real values and give approximate results.

Terms used with classes

Steps for grouping data

1. Find the smallest and largest values and calculate the range.
2. Choose 5 to 10 classes of equal width, so that the width is a convenient number and the classes together cover the whole range.
3. Write the class intervals so that they do not overlap and every value fits in exactly one class.
4. Tally each value into its class and count the tallies.
5. Add the frequencies and check the total against the number of observations.
6. Calculate the mid-point of each class and, if needed, the cumulative frequency.

Worked example

Example: The chloride concentrations (mg/L, to the nearest whole number) of 40 water samples were:

59, 25, 59, 59, 68, 50, 26, 41, 39, 53, 53, 56, 30, 62, 46, 38, 35, 30, 71, 59, 50, 46, 48, 41, 43, 40, 46, 60, 78, 22, 41, 31, 62, 51, 40, 31, 43, 64, 48, 66

The smallest value is 22 and the largest is 78, so the range is 56. Six classes of width 10 cover the data.

The frequencies add to 40, the number of samples. The modal class is 40-49. The values in the 20-29 class are 25, 26 and 22, which shows how the individual values are no longer visible once the data are grouped.