Statistical Diagrams

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Lesson notes

Stem & Leaf Diagrams

A stem-and-leaf diagram displays an ordered list of data using digits; two-digit numbers split into tens (stem) and units (leaf).
The stem is written vertically, leaves horizontally in order; include a key showing how values are formed (e.g., 1|8 means 18).
To find the median, cross out numbers from both ends until the middle is reached; remember to include the stem (e.g., median = 26, not 6).
The range is the difference between the highest and lowest values; the highest is at the end of the last stem.

Tally Charts & Frequency Tables

A tally chart uses vertical lines to count data; the fifth line is drawn diagonally across the previous four (e.g., ||||).
Convert tally marks to a frequency table by counting totals (e.g., |||| || = 5+2 = 7).
Frequency tables show how many times each item appears; they can be used to find the mode (most frequent item).

Bar Charts & Pictograms

A bar chart displays discrete data with bars of equal width separated by gaps; the vertical axis shows frequency, horizontal axis shows categories.
The mode is the category with the highest bar; for grouped data, the median can be found by converting to a table and using cumulative frequency.
A pictogram uses symbols to represent frequency; a key shows the value of one symbol (e.g., one shoe = 2 students).
Half or quarter symbols can represent fractions of the key value; pictograms have no axes.

Bar chart: shoe sizes in class 11A

Pie Charts

A pie chart is a circle divided into sectors showing relative proportions; each sector angle is (frequency/total) × 360°.
To draw, calculate each sector angle, draw a vertical radius, then use a protractor to mark angles in order.
If a pie chart says 'not to scale', use ratio and proportion (e.g., 30° = 15 people → 1° = 0.5 people).
The total angle is 360°, representing the total frequency; angles are proportional to frequencies.

Pie chart: favourite colours (30 students)

Reading & Interpreting Statistical Diagrams

Read titles, axis labels, and keys carefully; note units (e.g., thousands) and any extreme values (outliers).
Describe trends using numbers from the graph (e.g., 'temperature decreased from 12°C to 9°C over 3 days').
Be cautious about drawing conclusions: data may be unrepresentative (e.g., only one week) or biased.
Use mode, median, mean, or range to support explanations; write in full sentences copying exact wording.

Garden temperature at 7am each day

Comparing Statistical Diagrams

Compare trends (increases, decreases, peaks) and spread using range; use numbers from both data sets.
Calculate averages (mean, median, mode) to compare central tendency; relate calculations to context.
Comment on similarities and differences; note any outliers or unusual patterns.
State limitations: data may be from a specific period (e.g., opening week) and not representative of normal patterns.

Comparing books bought by male and female customers

Slides

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Practice questions

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1.In a stem-and-leaf diagram, what does the key '3|5' mean?
Easy
- A35
- B3.5
- C53
- D305
2.What is the mode of the data set: 2, 3, 3, 5, 7, 7, 7, 8?
Easy
- A3
- B5
- C7
- D8
3.A pictogram uses one symbol to represent 4 cars. How many symbols are needed to show 20 cars?
Easy
- A4
- B5
- C16
- D80
4.In a bar chart, the height of a bar shows the
Easy
- Afrequency
- Bcategory
- Ctotal
- Drange
5.What is the total angle in a pie chart?
Easy
- A180°
- B270°
- C360°
- D90°
6.A pie chart has a sector of 90° representing 30 people. How many people does the whole pie chart represent?
Medium
- A90
- B120
- C30
- D360
7.In a frequency table, the tally marks '|||| |||' represent what frequency?
Easy
- A7
- B8
- C9
- D5
8.The number of cars in a car park each day: 124, 130, 129, 116, 132, 120, 127, 107, 118, 114. What is the range?
Easy
- A25
- B23
- C24
- D26

Revision notes

## Statistical Diagrams - Quick Revision ### Stem & Leaf Diagrams - Split two-digit numbers into stem (tens) and leaf (units). - Write stems vertically, leaves horizontally in order. - **Key**: e.g., 1|8 means 18. - **Median**: middle value; cross out from ends until one or two remain; if two, find midpoint. ### Tally Charts & Frequency Tables - Tally: count in groups of five (||||). - Frequency table: total tallies for each category. ### Bar Charts & Pictograms - **Bar chart**: discrete data; bars separated, equal width; mode = highest bar. - **Pictogram**: uses symbols; key shows value per symbol; half/quarter symbols allowed. ### Pie Charts - Sector angle = (frequency / total) × 360°. - Total angle = 360°. - Use ratio: if angle a° represents value v, then 1° = v/a, total = (360 × v)/a. ### Reading & Interpreting Diagrams - Read titles, axes, keys, units. - Look for trends (increase, decrease, peak). - Use numbers from graph; write full sentences. - Consider if data is representative (e.g., small sample, biased). ### Comparing Diagrams - Compare averages (mean, median, mode) and spread (range). - Use numbers and context. - Comment on similarities/differences, outliers. ### Key Formulas - **Range** = highest value – lowest value. - **Mean** = sum of values / number of values. - **Median**: middle value when ordered. - **Mode**: most frequent value. - **Probability** = number of favourable outcomes / total outcomes.

Stem-and-leaf diagram: A simple way to display ordered data using digits; the stem is the tens digit and the leaf is the units digit.
Tally chart: A quick way to count data using groups of five vertical lines.
Frequency table: A table showing how many times each item appears.
Bar chart: A visual representation of discrete data using bars of equal width separated by gaps.
Pictogram: A diagram using symbols to represent data; a key shows the value of one symbol.
Pie chart: A circle divided into sectors representing categories; sector angle is proportional to frequency.
Mode: The most frequently occurring value.
Median: The middle value when data is ordered.
Range: The difference between the highest and lowest values.
Mean: The sum of all values divided by the number of values.

Flashcards

What is a stem-and-leaf diagram?

A stem-and-leaf diagram is a simple way to display an ordered list of data using digits. Two-digit numbers are split into a tens digit (the stem) and a units digit (the leaf).

In a stem-and-leaf diagram, how is the number 25 represented?

25 is represented as 2 | 5, where 2 is the stem (tens) and 5 is the leaf (units).

What is the purpose of the key in a stem-and-leaf diagram?

The key shows how values are formed from digits and should include units. For example, '2 | 5 represents 25 years old'.

How do you find the median from a stem-and-leaf diagram?

The median is the middle number. Since data values are already in order, cross out numbers from the beginning and end until they meet in the middle. If two numbers remain, find the midpoint between them.

What is a tally chart?

A tally chart is a quick way to count data. Each time you count an item, draw a vertical line. For the fifth count, draw a diagonal line through the four vertical lines (||||). Continue counting in sets of five.

What is a frequency table?

A frequency table counts how many times an item appears. Tally charts can be converted into frequency tables by finding the totals.

What is a bar chart?

A bar chart is a visual way to represent discrete data. The horizontal axis shows outcomes, the vertical axis shows frequency. Bars should be separated by gaps and have equal widths.

What is a pictogram?

A pictogram is an alternative to a bar chart where frequency is represented by symbols. A key shows the value of one symbol. Half and quarter symbols are often used.

In a pictogram, if one symbol represents 2 students, how many students does half a symbol represent?

Half a symbol represents 1 student.

What is a pie chart?

A pie chart is a circle divided into sectors that represent different categories. The sectors show the relative proportions of the categories, not the actual frequencies.

How do you calculate the sector angle for a category in a pie chart?

Multiply the fraction of the total frequency for that category by 360°. For example, if 11 out of 30 students chose red, the angle is (11/30) × 360° = 132°.

In a pie chart, what does 360° represent?

360° represents the total frequency.

If a sector of 30° represents 15 people, what does 1° represent?

1° represents 0.5 people (15 ÷ 30 = 0.5).

How do you find the mode from a bar chart?

The mode is the outcome with the highest bar (the tallest bar).

How do you find the total frequency from a bar chart?

Add up the heights (frequencies) of all the bars.

When comparing two data sets using statistical diagrams, what measures should you compare?

Compare averages (mean, median, mode) and the range (spread) of the data sets.

What does the range tell you about a data set?

The range tells you the spread of the data; it is the difference between the highest and lowest values.

What is an outlier in a statistical diagram?

An outlier is an extreme value that is much higher or lower than the rest of the data.

Why might a pie chart be labelled 'not to scale'?

It means you should use ratio and proportion methods to answer questions, not measure angles with a protractor.

What is a dual bar chart?

A dual bar chart has bars in pairs (side-by-side) for each outcome, used to compare two data sets.

How do you find the median from a bar chart?

Convert the bar chart data into a table, then use the method for finding the median from a frequency table.

What should you look for when interpreting a statistical diagram?

Read titles, axes labels, keys, and units. Look for trends, outliers, and use numbers from the graph to support conclusions.

Give one reason why data from a graph might not be representative.

The data set may be too small, biased, or only cover a specific time period (e.g., opening week) that is not typical.

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