Summary: Data Handling

Data

Data are pieces of information collected in the form of numbers, measurements or observations. When organised and presented properly, data become easier to understand and interpret. Collecting, recording and presenting data helps us to organise our experiences, draw conclusions and make decisions.

• Raw data is data in the original form as collected. It is often difficult to draw conclusions from raw data.
• Array means arranging the values of data in ascending or descending order.
• Range of a data set is the difference between the highest and the lowest values: range = highest value - lowest value.
• Tally marks are a simple way to record frequencies while collecting or organising data; they are later converted into numbers to prepare tables.
• Bar graph is a visual way to represent data using bars of uniform width. The length or height of each bar shows the frequency or value of the corresponding category.
• Double bar graph is used to compare two sets of observations side by side for the same categories.
• Mode in a bar graph corresponds to the category with the tallest bar (when bars represent frequencies).

Example: Ages of employees

Consider the ages (in years) of nine employees of an office:

40, 44, 26, 29, 33, 30, 24, 52, 28

Since these values are not ordered, this is raw data. Arrange them in ascending order to form an array:

• 24, 26, 28, 29, 30, 33, 40, 44, 52
• Range = 52 - 24 = 28 years.

Representative values (Measures of central tendency)

The three common measures of central tendency are mean, median and mode. They summarise a large set of observations by a single value that represents the central or typical value of the data.

Arithmetic mean (Mean)

Definition: The arithmetic mean of a data set is the sum of all observations divided by the number of observations.

Formula: Mean = (Sum of all observations) / (Number of observations)

Worked example - mean of the employee ages
Compute the mean of the ordered ages 24, 26, 28, 29, 30, 33, 40, 44, 52.
Sum of observations = 24 + 26 + 28 + 29 + 30 + 33 + 40 + 44 + 52.
Sum of observations = 306.
Number of observations = 9.
Mean = 306 ÷ 9.
Mean = 34.

Mode

Definition: The mode of a data set is the observation that occurs most frequently.

• If one value occurs more times than any other, that value is the mode.
• If all values occur exactly once or all occur equally often, then every value is a mode (we say the data are multimodal or have no unique mode for practical purposes).

For the example data: Each age appears only once, so the data has no unique mode.

Median

Definition: The median is the middle value of the data when the values are arranged in ascending or descending order so that half the observations lie on each side.

• If the number of observations is odd, the median is the middle value.
• If the number of observations is even, the median is the mean of the two middle values.

Worked example - median of the employee ages

The ordered ages are 24, 26, 28, 29, 30, 33, 40, 44, 52.

There are nine observations (an odd number), so the median is the middle (5th) value.

Median = 30.

Empirical relation between mean, median and mode

A commonly used empirical relation is:
3 × Median = Mode + 2 × Mean

This relation is an approximation that holds reasonably well for moderately skewed distributions; it provides a way to estimate one measure if the other two are known.

Organising and representing data

• Tally and frequency table: Use tally marks while counting occurrences; convert tallies into numeric frequencies and present them in a table for clarity.
• Bar graphs: Use bars of uniform width. The height (or length) of each bar represents frequency or amount. Label axes and give a clear title.
• Double bar graphs: Use two bars for each category (side by side) to compare two related data sets for the same categories; use a legend to distinguish the two sets.
• Choosing scales: Select an appropriate scale on the axis so that bars fit and differences are easy to compare.

Probability

Chance describes how likely an event is to happen. The probability of an event is a number between 0 and 1 that quantifies this likelihood.
Probability of an event = (Number of favourable outcomes) / (Total number of outcomes in the experiment)

• The probability of a certain event is 1.
• The probability of an impossible event is 0.

Simple examples

Example 1: Tossing a fair coin. The sample space is {Head, Tail}.
Number of favourable outcomes for getting a Head = 1.
Total number of outcomes = 2.
Probability of getting a Head = 1 ÷ 2 = 1/2.

Example 2: Rolling a fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}.
Probability of getting a 4 = 1 ÷ 6 = 1/6.

Applications and summary

• Measures of central tendency (mean, median, mode) summarise data and help compare different data sets.
• Range gives a simple measure of how spread out the data are; combined with mean and median it helps to understand distribution shape.
• Graphical representations such as bar graphs and double bar graphs make comparisons easy and convey information quickly.
• Probability helps us make predictions about uncertain events and is the basis for risk assessment in daily life and science.

If you practise computing mean, median and mode for different types of data and draw bar graphs from frequency tables, you will gain confidence in handling and interpreting data.