Tag: statistics

Bar Graphs, Mean, Median and Mode: How to Draw, Read and Average Data

Aven BrisailLeave a comment

In Tally Charts, Pictographs and Block Graphs you organised data and found the mode. Now you will draw bar graphs with a sensible scale, read information from them, and describe a set of data with three kinds of average: the mode, the median and the mean. Averages are used everywhere: a cricketer’s batting average, the average rainfall in a month, or the typical price of a house.

Drawing a bar graph

  1. Draw two axes with a ruler. Put the categories (days, colours, sports…) along the horizontal axis and the frequency (how many) up the vertical axis.
  2. Choose a scale that fits the biggest value and goes up in equal steps – for example 1 square = 2, 5 or 10. Label the scale starting at 0.
  3. Draw bars of equal width with equal gaps between them. Each bar’s height is its frequency.
  4. Label both axes and give the graph a title.
Bar graph of drinks sold at the school tuck shop: Monday 35, Tuesday 20, Wednesday 45, Thursday 30, Friday 50.01020304050MonTueWedThuFriDayDrinks soldDrinks sold at the school tuck shop
Figure 1 – A bar graph with a scale of 10 drinks per grid line. Values between grid lines (35, 45) are read halfway.

Reading the graph: the most drinks were sold on Friday (50) and the fewest on Tuesday (20) – a difference of 30. Altogether 35 + 20 + 45 + 30 + 50 = 180 drinks were sold, a mean of 180 ÷ 5 = 36 per day.

Mode, median and mean

  • The mode is the value that occurs most often. There can be more than one mode, or none.
  • The median is the middle value when the data are put in order. With an even number of values, it is halfway between the two middle values.
  • The mean is the total of all the values divided by how many values there are. It is what each value would be if the total were shared out equally.

TRY ITMean, median and mode calculator

Type a set of whole numbers. The bar graph shows how often each value appears, and the mean, median and mode are worked out step by step.

Bar graph of the data

Worked examples

Example 1: All three averages

Question: Find the mode, median and mean of 4, 7, 3, 7, 9, 2, 7, 5.

  1. In order: 2, 3, 4, 5, 7, 7, 7, 9 (8 values).
  2. Mode: 7 appears three times, so the mode is 7.
  3. Median: the middle values are the 4th and 5th, 5 and 7, so the median is (5 + 7) ÷ 2 = 6.
  4. Mean: total = 44, and 44 ÷ 8 = 512 (5.5).

Answer: Mode 7, median 6, mean 512.

Example 2: Median of an even set

Question: Find the median of 12, 15, 11, 18, 14, 16.

  1. In order: 11, 12, 14, 15, 16, 18.
  2. The two middle values are 14 and 15.
  3. Halfway between them: (14 + 15) ÷ 2 = 14.5.

Answer: 14.5.

Example 3: A missing value

Question: The mean of five test marks is 8. Four of them are 6, 9, 7 and 10. Find the fifth.

  1. Mean 8 for 5 marks means the total is 5 × 8 = 40.
  2. The four known marks total 6 + 9 + 7 + 10 = 32.
  3. 40 − 32 = 8.

Answer: The fifth mark is 8.

Example 4: Which average?

Question: Five friends earn $20, $22, $21, $23 and $95 for a job. Which average best describes a typical amount?

  1. Mean = 181 ÷ 5 = $36.20 – pulled up by the one big value of $95.
  2. Median = $22 (the middle value).

Answer: The median ($22) is more typical; one unusual value distorts the mean.

Which average to use? The mode suits categories (favourite colour) or the most common size (a shop ordering shoes). The median is good when there are a few very large or very small values. The mean uses every value and is best when the data have no extreme values.

Common mistakes

  • Finding the median without ordering the data first.
  • Giving the frequency instead of the value as the mode: if 7 appears 3 times, the mode is 7, not 3.
  • Dividing by the wrong number for the mean – count the values carefully.
  • Bars of different widths, or a scale with uneven steps or not starting at 0.

Practice questions

Try these yourself first, then open each answer to check your working.

1. Find the mean, median and mode of 3, 8, 5, 6, 8.

Show answer

Mean 30 ÷ 5 = 6; median 6 (order 3, 5, 6, 8, 8); mode 8.

2. Find the mean and median of 10, 4, 7, 9.

Show answer

Mean 30 ÷ 4 = 7.5; median (7 + 9) ÷ 2 = 8.

3. The mean of four numbers is 15. Three of them are 12, 18 and 14. Find the fourth.

Show answer

4 × 15 = 60; 60 − 44 = 16.

4. The mean age of 6 children is 24 months. A 31-month-old joins them. What is the new mean?

Show answer

Total 144 + 31 = 175; 175 ÷ 7 = 25 months.

5. Goals in 9 matches: 2, 0, 1, 3, 1, 0, 4, 1, 2. Find the mode, median and mean.

Show answer

Mode 1; median 1 (order 0, 0, 1, 1, 1, 2, 2, 3, 4); mean 14 ÷ 9 = 159 ≈ 1.6.

6. On a bar graph, one square represents 5 people. How many people do bars of 8 squares and 13 squares show?

Show answer

40 and 65.

Quick check

1. Which average is the middle value?

Show answer

The median.

2. Can a set of data have two modes?

Show answer

Yes.

3. Should the bars of a bar graph have equal widths?

Show answer

Yes, with equal gaps.

Next: Mass and Weight: Grams, Kilograms and Converting Units

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 2, Strand 1.3 Statistics and Probability, Topic 1.3.2 Statistics – Learning Outcomes 1.3.2.1 (constructing bar graphs with suitable scales), 1.3.2.2 (interpreting bar graphs) and 1.3.2.3 (solving problems involving the mode, median and mean).

© 2026 Vedesh Kungebeharry. All rights reserved.

Tally Charts, Frequency Tables, Pictographs and Block Graphs (and How to Find the Mode)

Aven BrisailLeave a comment

Statistics is the part of mathematics that deals with collecting, organising, displaying and interpreting data (information, usually numbers or categories). Schools, businesses and governments use it to make decisions: which snacks a school café should sell, which bus route needs more buses, how many vaccines a country needs. In this lesson you will follow the full statistics process – asking a question, collecting discrete data, using a tally chart and frequency table, drawing pictographs and block graphs, drawing conclusions, and finding the mode.

Step 1: Ask a question that data can answer

A good statistical question has an answer that varies and can be counted. “What is your favourite fruit?” is a good survey question because people give different answers that can be sorted into categories. “Do you like fruit?” gives much less useful information. A bigger investigation might ask: “What healthy snacks should our school café sell, and why?” – and then use survey questions to find out.

Step 2: Collect discrete data

Discrete data can only take separate, countable values – usually whole numbers or categories. Examples: the number of brothers and sisters you have (0, 1, 2, …), the number of goals scored, or favourite sport. (Data such as height or time, which can take any value on a scale, is called continuous – you will meet it in later forms.) Ways to collect data include:

  • Surveys and questionnaires – asking people questions.
  • Observation and counting – for example, counting cars passing the school gate.
  • Interviews – talking to people one at a time.
  • Experiments – for example, rolling a die 50 times.
  • Existing records – newspapers, books, websites and databases.

Step 3: Organise the data – tally chart and frequency table

Here are the favourite sports of 30 students, in the order they answered:

Football, Cricket, Netball, Football, Basketball, Cricket, Football, Athletics, Netball, Football, Cricket, Basketball, Football, Netball, Cricket, Athletics, Football, Basketball, Netball, Cricket, Football, Netball, Basketball, Cricket, Football, Athletics, Netball, Basketball, Cricket, Football.

Go through the list once. For every answer, make a tally mark (|) next to that sport. Every fifth mark is drawn across the previous four, so the marks are in easy-to-count bundles of five. The frequency is how many times each answer occurs.

Hand-drawn tally marks in bundles of five showing 11, 12 and 13
Figure 1 – Tally marks for 11, 12 and 13: two bundles of five, then single strokes. Image: 293u8h1, CC0, via Wikimedia Commons.
SportTallyFrequency
Football|||| ||||9
Cricket|||| ||7
Netball|||| |6
Basketball||||5
Athletics|||3
Total30
Table 1 – Tally chart and frequency table of favourite sports.

Always check that the frequencies add up to the number of answers: 9 + 7 + 6 + 5 + 3 = 30 ✔

Step 4: Display the data

Pictographs

A pictograph (pictogram) uses a small picture or symbol to stand for a number of items. It must have a key. With 30 students, one symbol for every 2 students keeps the graph short; half a symbol stands for 1 student.

Football
Cricket
Netball
Basketball
Athletics
Figure 2 – Pictograph of favourite sports. Key: = 2 students (a half symbol = 1 student).

Football has 9 students: 9 ÷ 2 = 4 remainder 1, so it gets 4 whole symbols and a half symbol.

Block graphs

A block graph stacks one block (a square) for each item or for a fixed number of items. Blocks are the same size and sit on a common baseline, and the vertical axis is numbered from 0 with an even scale. Give the graph a title and label both axes. A block graph is a stepping stone to the bar graphs you will draw in Term 2. Try building one in the interactive below.

TRY ITTally chart, frequency table and block graph

Collect your own data: press a button for each answer. The tally marks, frequency table, block graph and mode update as you go.

Survey question: “What is your favourite fruit?” Press a button each time someone answers.

FruitTallyFrequency
Total0

Block graph (1 block = 1 person):

Block graph

Step 5: Interpret and draw conclusions

Reading a graph means answering questions such as: Which category is biggest or smallest? How many more chose one option than another? What fraction of the total chose a category? Drawing a conclusion means saying what the data tells us about the original question – and noticing its limits.

  • Football is the most popular sport (9 students) and athletics the least popular (3 students).
  • 9 − 3 = 6 more students chose football than athletics.
  • Conclusion: in this group, a football club would probably attract the most members. But only 30 students were asked, so the result might be different for the whole school.

The mode

The mode is the value (or category) that occurs most often – the one with the highest frequency. In Table 1 the mode is football. A set of data can have more than one mode if two values tie, or no useful mode if every value occurs equally often.

Worked example: The mode from a frequency table

Question: 25 students were asked how many brothers and sisters they have. Find the mode and answer the questions.

  1. Table: 0 siblings – 4 students; 1 sibling – 9 students; 2 siblings – 7 students; 3 siblings – 3 students; 4 siblings – 2 students.
  2. The highest frequency is 9, and it belongs to the value 1 sibling.
  3. Check the total: 4 + 9 + 7 + 3 + 2 = 25 ✔
  4. Students with 2 or more siblings: 7 + 3 + 2 = 12.

Answer: The mode is 1 sibling (not 9 – the 9 is the frequency, not the value).

Common mistakes

  • Giving the frequency as the mode. The mode is the value with the highest frequency (“1 sibling”, “football”), not the frequency itself.
  • Ignoring the pictograph key. If one symbol stands for 4 people, 3 symbols mean 12 people, not 3.
  • Drawing the fifth tally mark as a fifth upright stroke. The fifth mark crosses the other four, making a bundle you can count by fives.
  • Uneven blocks or scales. Every block must be the same size, and the numbers on the axis must go up in equal steps from 0.
  • Not checking the total. If the frequencies do not add up to the number of answers, a tally was missed or doubled.

Practice questions

Try these yourself first, then open each answer to check your working.

1. How do you show a frequency of 13 with tally marks?

Show answer

Two bundles of five (four strokes with a fifth across them), then three single strokes: 5 + 5 + 3 = 13.

2. These are the colours of 15 cars that passed a school gate (R = red, B = blue, W = white, S = silver): R, B, W, W, S, R, W, B, W, S, W, R, B, W, S. Make a frequency table and find the mode.

Show answer

Red 3, Blue 3, White 6, Silver 3 (total 15). The mode is white.

3. In a pictograph, one book symbol stands for 4 books. A row shows 3½ symbols. How many books is that?

Show answer

3½ × 4 = 14 books.

4. Using the siblings table in the worked example, how many students were asked, and how many have no brothers or sisters?

Show answer

25 students were asked; 4 have no siblings.

5. Why is “Which bus route do you use to get to school?” a better survey question than “Do you take the bus?” for planning bus routes?

Show answer

It gives the actual routes (categories that can be counted), so planners can see which routes are used most. “Do you take the bus?” only gives yes or no.

6. In a block graph each block stands for 5 people. The “Mango” column has 9 blocks. How many people chose mango?

Show answer

9 × 5 = 45 people.

Quick check

1. What is discrete data?

Show answer

Data that can only take separate, countable values, such as whole numbers or categories.

2. What does the frequency column in a table tell you?

Show answer

How many times each value or category occurs.

3. What must every pictograph have so that it can be read?

Show answer

A key showing how many items each symbol stands for.

4. What is the mode?

Show answer

The value or category that occurs most often (highest frequency).

Next: Introduction to Algebra: Variables, Constants, Expressions and Substitution

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.3 Statistics and Probability, Topic 1.3.1 Statistics – Learning Outcomes 1.3.1.1 (formulating a problem or questions that data can answer), 1.3.1.2 (collecting discrete data), 1.3.1.3 (tallying ungrouped discrete data into a frequency table), 1.3.1.4 (constructing pictographs and block graphs with suitable scales), 1.3.1.5 (interpreting them), 1.3.1.6 (drawing conclusions from them) and 1.3.1.7 (finding the mode from a frequency table).

Attributions to media used in this post:

Tally Marks.png – 293u8h1 – CC0 (via Wikimedia Commons)

© 2026 Vedesh Kungebeharry. All rights reserved.