statistics - chapter 2: processing and representing data

What are you expected to do with tables in exams?

Extract and interpret information.

What skills are needed when working with tables?

• Identify values

• Calculate totals

• Calculate differences

• Calculate percentages

• Describe trends

• Explain inconsistencies

What is a database?

A table containing a collection of data, usually secondary data that is easily accessible online.

Why don’t percentages in tables always add up to 100%?

Due to rounding errors.

Why should you think about real-life context in tables?

Because the data represents real-world situations, so trends need real-life explanations.

What is a two-way table?

A table showing data in two categories (bivariate data).

How do you find missing values in a two-way table?

Start with the row or column that has only one missing value

What must totals in a two-way table do?

Row totals and column totals must match overall.

How should you compare data in two-way tables?

• Compare rows and columns

• Compare individual cells

What is a pictogram?

A chart using symbols/pictures to represent data.

What must a pictogram always include?

A key showing the value of each symbol.

Rules for drawing pictograms?

• Symbols must be the same size

• Values should divide easily

• You can use fractions of symbols

• Equal spacing in each row

• Include a key

Key features of a simple bar chart?

• Equal width bars

• Equal gaps

• Frequency on the y-axis

What is a vertical line graph?

Like a bar chart but uses lines instead of bars.

What are multiple bar charts used for?

Comparing two or more data sets.

How are categories shown?

Using different coloured bars and a key

What is a composite bar chart?

A bar split into sections for different categories

What are they used for?

Comparing data across time (days, years, etc.).

How do you find the value of a section?

Subtract the lower value from the upper value.

What should you NOT do when reading composite bars

Don’t just read from the y-axis (except total or bottom section).

Why use a stem and leaf diagram?

• Keeps all original data

• Shows distribution shape

What does it show about data?

Whether values cluster at the start, middle, or end.

How is data split?

• Stem = first digit(s)

• Leaf = last digit

Rules for stem and leaf diagrams?

• Leaves are single digits

• No commas

• Leaves in ascending order

• Include a key

what is back to back stem and leaf diagram used for?

Comparing two data sets.

Key features of back to back stem and leaf

• Shared stem

• Values closest to stem are smallest

• Two separate keys

What is a pie chart

Shows how data is divided into categories

What does each sector represent?

A proportion of the total

What is the total angle of a pie chart?

360°

What does the area represent?

Total frequency.

steps to drawing pie chart

Step 1: Find total frequency.

Step 2: Calculate angle per frequency: 360 ÷ total frequency

Step 3: Multiply by each category frequency.

Step 4: Check angles add to 360°.

Step 5: Draw chart.

Step 6: Label sectors.

what are comparative pie charts used for?

Comparing data sets of different sizes.

Why must circles be different sizes for comparative pie charts?

Same size charts can be misleading.

What must match between comparative pie charts?

Area ratio = frequency ratio.

finding radius for comparative pie charts

Step 1: Divide areas (frequency ratio).

Step 2: Square root result.

Step 3: Multiply by original radius.

Formula for comparative pie chart radius?

r₂ = r₁ × √(F₂ / F₁)

Key interpretations of comparative pie charts

• Bigger pie = larger total frequency

• Same angles ≠ same frequency (depends on size)

What do population pyramids show?

Age distribution of a population.

What units are used in pop pyramids?

Numbers or percentages.

What are pop pyramids used for?

Comparing genders or regions.

What does a pyramid shape (wide bottom) mean?

• More young people

• High birth rate / low life expectancy

What does a rectangular pyramid shape mean?

• Even age distribution

• Lower birth/death rates

• Increasing life expectancy

What does an upside-down pyramid mean?

• More older people

• Low birth rate

• High life expectancy

Possible reasons for an ageing population?

• Low birth/death rates

• Longer life expectancy

• Retirement migration

What is a choropleth map?

A map that uses shading (colour by numbers) to represent data across regions.

How are regions shown on a choropleth map

The area is split into regions, each shaded differently

What does darker shading represent in chloropleth maps?

A higher frequency or value.

What is the purpose of the key in chloropleth maps?

To show what each shading level represents.

How do you interpret a choropleth map?

• The darkest area = highest proportion/percentage

• Use the key to read exact values

What is cumulative frequency?

A running total of frequencies.

How do you calculate CF for a class interval?

Add the frequency of that class to the CF of the previous class.

What do you use on the x-axis - CF ?

Upper class boundaries.

When do you use CF step polygons?

For discrete data.

How are CF step polygons drawn?

• Plot upper bounds

• Join points with straight lines

• Go across, then up

When do you use CF curves?

For grouped continuous data.

How are CF curves drawn?

• Plot upper bounds

• Join points with a smooth curve

How do you find the median from a CF graph?

• Divide total frequency by 2

• Find this on the y-axis

• Draw a horizontal line to the curve

• Drop down to x-axis to read value

How do you find IQR?

• Find 25% and 75% of total frequency

• Locate them on y-axis

• Draw lines to curve

• Read values from x-axis

• Subtract (Q3 − Q1)

How do you estimate values greater than a number?

• Draw vertical line from x-value to curve

• Read y-value

• Subtract from total frequency

What do histograms represent?

Continuous data from grouped frequency tables

Do histograms have gaps?

No.

What do axes represent - histograms - equal class widths

• x-axis = data

• y-axis = frequency

What do histograms look like?

Bar charts without gaps

What does bar area represent?

Frequency

What is on the y-axis?- unequal class widths

Frequency density

What does frequency density show?

The concentration of data in each interval

Formula for frequency density?

Frequency Density= frequency/class width

Rearranged formula from frequency density

Frequency=Frequency Density × Class Width

Steps to draw a histogram?

1. Calculate class widths

2. Calculate frequency density (FD = F ÷ CW)

3. Choose scale for y-axis (frequency density)

4. Draw bars (no gaps)

How do you estimate frequencies?

• Identify relevant bars

• Use FD × CW = F

• Add frequencies

What must you be careful of? - estimating from histograms

Intervals that don’t cover the whole bar

What must be the same when comparing?

• Class intervals

• Frequency density scale

What should you describe? - comparing histograms

Shape of distribution and what it shows

What are the three types of distribution shapes?

• Positive skew

• Symmetrical

• Negative skew

What is positive skew?

• Data mostly at lower values

• Tail extends right

What is symmetrical distribution?

• No skew

• Even spread

What is negative skew?

• Data mostly at higher values

• Tail extends left

What is a frequency polygon?

A line graph similar to a histogram but without bars.

What points are used? - freq polygons

Midpoints of class intervals

How are they drawn - freq polygons

Plot points and join with straight lines

Common mistakes in frequency polygons?

• Not using midpoints

• Joining at bottom

• Using curves instead of straight lines

Why can diagrams be misleading?

Due to shape, axes, or scales.

Issues with pictograms?

Symbols must be same size and have a key

Issues with 3D charts?

Distort values

Issues with colours?

Some colours exaggerate importance

Issues with lines?

Thick lines make reading harder

What axis issues can mislead?

• Doesn’t start at zero

• Missing values

• Uneven scaling

• No labels

• No key