Data Analysis Study Notes

Key Concepts

• Data analysis is the process of collecting, organising, displaying and interpreting data so that patterns and conclusions can be seen clearly.

• In Lower Secondary Mathematics, data is often collected from:

  • experiments
  • surveys
  • observations
  • measurements

• Data means information collected during an investigation.

  • Example: temperatures measured every hour, number of seeds germinating, or reaction times of students.

• Data can be shown in different forms. Common statistical diagrams include:

  • dot diagrams
  • stem-and-leaf diagrams
  • histograms

• Different diagrams are useful for different purposes:

  • Dot diagrams are good for small sets of data.
  • Stem-and-leaf diagrams help you organise data while keeping the original values.
  • Histograms are used for grouped continuous data and help show the shape of the distribution.

• Measures of central value help us describe a data set using one representative number:

  • mean
  • median
  • mode

• When comparing data sets, we should not only look at the average value, but also:

  • how spread out the data is
  • whether the data is clustered
  • whether there are unusually high or low values
  • whether one graph is more symmetrical or skewed than another

• Interpreting statistical diagrams means reading information from graphs or charts and explaining what the data shows.

Dot Diagrams

• A dot diagram displays data values using dots placed above a number line.
• Each dot represents one piece of data.
• If several values are the same, the dots are stacked vertically.

What a dot diagram shows

• the individual values in the data set
• the most common value
• the spread of the data
• clusters and gaps
• any unusual values

When to use it

• for small data sets
• when exact values are important
• when the values are not too many

Stem-and-Leaf Diagrams

• A stem-and-leaf diagram shows data by splitting each number into:

  • a stem: the leading digit or digits
  • a leaf: the last digit

• Example:

  • 27 can be split into stem 2 and leaf 7
  • 43 can be split into stem 4 and leaf 3

• The stem-and-leaf diagram keeps the original data values, unlike some grouped graphs.

Features

• data must be arranged in ascending order
• leaves are written from smallest to largest within each stem
• a key must always be included to show how to read the numbers

What it helps you see

• the distribution of values
• the smallest and largest value
• the mode
• the median
• clusters and gaps

Histograms

• A histogram is a graph used to show the frequency distribution of continuous data grouped into class intervals.

• Continuous data is data that can take any value within a range.

  • Example: height, mass, time, temperature

• In a histogram:

  • the horizontal axis shows the class intervals
  • the vertical axis shows the frequency or sometimes frequency density
  • bars touch each other because continuous data has no gaps between intervals

Important features

• bars are drawn with no spaces between them
• the width of each bar represents the class interval
• the height represents frequency if intervals are equal
• if class widths are unequal, the height represents frequency density

What a histogram shows

• how data is distributed
• which intervals contain the most values
• whether data is spread evenly or clustered
• the general shape of the data

Mean, Median and Mode

These are three common ways to describe the centre of a data set.

Mean

• The mean is the average value.
• It is calculated by:

• The mean uses all the values in the data set.
• It can be affected by extreme values.

Median

• The median is the middle value when the data is arranged in order.

• If there is an odd number of values:

  • the median is the middle value

• If there is an even number of values:

  • the median is the average of the two middle values

• The median is less affected by extreme values than the mean.

Mode

• The mode is the value that occurs most often.
• A data set may have:
  • one mode
  • more than one mode
  • no mode if no value repeats

When to use each average

• Exam questions may ask: “Which average best represents the data? Explain your answer.”
  • If there is an outlier, say: “The median is a better representative because the mean is pulled up/down by the extreme value.”
  • If data is categorical, say: “The mode is the only appropriate average for categorical data.”

Comparing Data Sets

When comparing two or more sets of data, look at:

• centre
  • compare mean, median or mode
• spread
  • see whether the values are close together or widely spread out
• range
  • the difference between the largest and smallest values
• shape
  • whether the data is symmetrical, clustered, or skewed
• consistency
  • a smaller spread usually means more consistent results

Range

• A small range means the data is less spread out.
• A large range means the data is more spread out.

Why comparison matters in Science

• to decide which result is more reliable
• to tell which condition gives higher or lower values
• to identify variation in repeated measurements
• to compare groups in experiments

Interpreting Statistical Diagrams

To interpret a statistical diagram, you should be able to:

• read exact values correctly
• identify the highest and lowest values
• state the most common value or interval
• describe patterns
• compare groups
• draw reasonable conclusions based on the data

Useful observations to make

• “Most of the values are between … and …”
• “The highest frequency is in the interval …”
• “The data is spread from … to …”
• “There is a cluster around …”
• “There is a gap between … and …”
• “Set A has a higher mean than Set B.”
• “Set B is more consistent because its values are less spread out.”

Be careful

• conclusions must match the data shown
• do not guess values not supported by the graph
• always include units where needed

Important Definitions

• Data: information collected from observations, measurements, experiments or surveys.

• Data analysis: the process of organising, presenting and interpreting data to find patterns and draw conclusions.

• Frequency: the number of times a value occurs.

• Distribution: the way data values are spread out over a range.

• Dot diagram: a statistical diagram in which each data value is represented by a dot above a number line.

• Stem-and-leaf diagram: a diagram that organises data by separating each value into a stem and a leaf.

• Stem: the leading digit or digits of a number in a stem-and-leaf diagram.

• Leaf: the final digit of a number in a stem-and-leaf diagram.

• Key: a note used in a stem-and-leaf diagram to show how to read the stems and leaves.

• Histogram: a graph of grouped continuous data in which touching bars represent frequencies in class intervals.

• Continuous data: data that can take any value within a given range.

• Class interval: a range of values grouped together in a table or histogram.

• Mean: the sum of all data values divided by the number of values.

• Median: the middle value in an ordered set of data.

• Mode: the value that appears most frequently in a data set.

• Range: the difference between the highest and lowest values in a data set.

• Cluster: a group of values close together.

• Gap: a part of the data range where there are no values.

• Outlier: a value that is much higher or lower than the others.

• Interpret: to explain what the data shows.

Worked Examples

Worked Example 1: Finding Mean, Median and Mode

A group of students recorded the number of hours they slept: 8, 7, 6, 8, 9, 7, 8

Step 1: Arrange the data in order

6, 7, 7, 8, 8, 8, 9

Step 2: Find the mean

Add all the values:

Number of values = 7

Mean = 7.57 hours
(You may round appropriately, for example to 7.6 hours.)

Step 3: Find the median

There are 7 values, so the median is the 4th value.

Median = 8 hours

Step 4: Find the mode

The value that appears most often is 8.

Mode = 8 hours

Final answers

• Mean = 7.57 hours
• Median = 8 hours
• Mode = 8 hours

Worked Example 2: Constructing a Stem-and-Leaf Diagram

The masses of 10 bags of rice are: 21, 25, 24, 27, 31, 29, 22, 24, 33, 28

Step 1: Arrange the data in order

21, 22, 24, 24, 25, 27, 28, 29, 31, 33

Step 2: Identify stems and leaves

• 21 → stem 2, leaf 1
• 22 → stem 2, leaf 2
• 24 → stem 2, leaf 4
• 24 → stem 2, leaf 4
• 25 → stem 2, leaf 5
• 27 → stem 2, leaf 7
• 28 → stem 2, leaf 8
• 29 → stem 2, leaf 9
• 31 → stem 3, leaf 1
• 33 → stem 3, leaf 3

Step 3: Draw the diagram

Stem | Leaf
 2   | 1 2 4 4 5 7 8 9
 3   | 1 3

Key: 2 | 1 = 21

Step 4: Interpret the data

• Lowest mass = 21
• Highest mass = 33
• Mode = 24
• Most values are in the 20s

Worked Example 3: Comparing Two Data Sets

Two groups of students measured the time taken to complete a task.

• Group A: 12, 13, 14, 15, 16
• Group B: 10, 13, 14, 15, 18

Step 1: Find the mean for Group A

Mean of Group A = 14

Step 2: Find the mean for Group B

Mean of Group B = 14

Step 3: Find the range for each group

For Group A:

Range of Group A = 4

For Group B:

Range of Group B = 8

Step 4: Compare the groups

• Both groups have the same mean: 14
• Group A has a smaller range
• Group A is more consistent
• Group B is more spread out

Final conclusion

Although both groups have the same average time, Group A is more consistent because its data has a smaller range.

Worked Example 4: Reading a Histogram

A histogram shows the heights of 30 students:

Questions:

1. How many students are between 155–165 cm tall?

• Add the frequencies for the two intervals:
• 8 + 12 = 20 students

2. What is the modal class?

• The modal class is the interval with the highest frequency.
• The 160–165 cm interval has frequency 12, which is the highest.
• Modal class = 160–165 cm

3. What fraction of students are 165 cm or taller?

• Students in 165–170 cm: 5
• Students in 170–175 cm: 1
• Total at 165 cm or taller: 5 + 1 = 6
• Fraction = 6/30 = 1/5

Reminder: In a histogram, the bars touch because the data is continuous — there are no gaps between class intervals.

Common Mistakes to Avoid

• Forgetting to arrange data in ascending order before finding the median.
• Choosing the wrong middle value for the median.
• Forgetting that for an even number of values, the median is the average of the two middle values.
• Adding wrongly when calculating the mean.
• Dividing by the wrong number when finding the mean.
• Confusing mode with median.
• Forgetting to include a key in a stem-and-leaf diagram.
• Writing leaves in random order instead of ascending order.
• Leaving gaps between bars in a histogram.
• Drawing a bar chart instead of a histogram.
• Using unequal scales without noticing.
• Reading the axes wrongly.
• Ignoring units such as cm, g, s or °C.
• Comparing only averages and not spread.
• Making conclusions that are not supported by the data.
• Ignoring outliers that may affect the mean.

Exam Tips

• Always state the unit in your answer if the data has one.

  • Example: 8 cm, 15 s, 27 °C

• When asked to compare data sets, mention both centre and spread.

  • Good phrase: “Set A has a higher mean, but Set B is more consistent because it has a smaller range.”

• For median questions:

  • first write the data in order
  • then clearly identify the middle value

• For mean questions:

  • show the total and the number of values
  • this earns method marks in working questions

• For stem-and-leaf diagrams:

  • write stems on the left
  • leaves on the right
  • include a key
  • arrange leaves in ascending order

• For histogram questions:

  • bars must touch
  • label both axes
  • use a suitable scale

• When interpreting a graph, use data from the diagram in your statement.

  • Example: “The highest frequency is in the 20–29 interval.”

• Avoid vague statements like:

  • “It is better”
  • “It is bigger”
  • Instead say:
    • “It has a higher mean”
    • “It has a smaller range”
    • “It shows greater variation”

• If there is an unusual value, mention it.

  • Example: “There is an outlier at 35 cm, which may affect the mean.”

• In Science, when comparing repeated measurements, use:

  • more consistent
  • less spread out
  • greater variation
  • more reliable

Quick Summary

• Data analysis means organising, presenting and interpreting data.
• A dot diagram uses dots above a number line to show individual values.
• A stem-and-leaf diagram keeps the original data values and must include a key.
• In a stem-and-leaf diagram, leaves must be arranged in ascending order.
• A histogram is used for continuous data grouped into class intervals.
• In a histogram, bars touch because the data is continuous.
• Mean = total of values ÷ number of values.
• Median = middle value in ordered data.
• Mode = most frequent value.
• Range = highest value − lowest value.
• To compare data sets, look at average values and spread.
• When interpreting diagrams, use exact data and include units in your answers.