Why are the bars on a bar chart drawn with gaps between them?

Because a bar chart shows discrete data, where each bar is a separate category rather than part of a continuous scale. The gaps make it clear that there are no values in between.

True or False? All the bars on a bar chart must be drawn with the same width.

True. Bars on a bar chart are always drawn with equal widths, so that the height on its own represents the frequency. If the widths varied, a taller bar would no longer reliably mean a larger frequency.

When is a vertical line graph a better choice than a bar chart?

When the data is numerical and there are a lot of different values to show, such as test scores out of 20. Thin lines fit where wide bars would be crowded, so the spread of the data and any unusual values stand out clearly.

What is the difference between a dual bar chart and a bar-line chart ?

A dual bar chart shows two data sets measuring the same variable, so the bars sit side by side on one shared scale. A bar-line chart shows two different variables, one as bars and one as a line, so each can have its own scale: monthly rainfall as bars with monthly temperature as a line.

A pictogram represents frequencies using symbols instead of bars, with no axes at all. A key states what one symbol is worth, and part-symbols stand for part-values, so if one symbol means 2 people then half a symbol means 1 person.

What is a composite bar chart , and what does the height of one coloured section show?

A composite bar chart stacks the categories within each bar, so the overall height of a bar gives the total frequency. A coloured section shows the frequency for that category alone, read as the difference between the two ends of the section rather than as the value at its top.

Why can a composite bar chart tell you something a plain bar chart of the same totals cannot?

Because it shows the breakdown within each bar as well as the total. A time slot can have the smallest total traffic of the day and still contain the largest number of lorries, and only the composite chart reveals that.

Why does a pie chart show proportions rather than actual amounts?

Because the whole circle always represents the whole data set, whatever its size, so a sector's angle only tells you what share of the total it is. Without being told the total frequency you cannot turn any sector back into a number of items.

True or False? A pie chart labelled 'not drawn accurately' can still have its angles measured with a protractor.

False. If a diagram is not drawn accurately then the angles on the page are not the true ones, so measuring them tells you nothing. Work the angles out from the frequencies instead, or use an angle the diagram states.

What makes comparative pie charts different from two ordinary pie charts?

Their areas are drawn in the same ratio as the total frequencies of the two data sets, so a bigger circle really does mean more data. Two ordinary pie charts drawn the same size make data sets of very different sizes look equal.

Why must a stem-and-leaf diagram always have a key?

Because the same diagram could stand for very different numbers. A stem of 3 with a leaf of 6 could mean 36 or 3.6, and only the key says which.

Define stem-and-leaf diagram .

A stem-and-leaf diagram splits each value into a stem , its leading digits, and a leaf , its last digit, putting all the values that share a stem on one row. It orders the data and groups it into classes while still showing every original value.

In a stem-and-leaf diagram the median turns out to be a leaf of 1 on the stem of 2. What is the median?

The median is 21 , not 1. A leaf gives only one digit, so you must put the value back together with its stem before writing it down, which is a common place to lose marks.

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

Look for the stem with the most leaves. Each stem is a class interval, so the longest row of leaves is the class holding the most data.

What is a back-to-back stem-and-leaf diagram used for?

Comparing two related sets of data, such as boys against girls, on one diagram instead of two. The two sets share a single central column of stems, with one set's leaves running out to the left and the other's to the right.

True or False? On a back-to-back stem-and-leaf diagram, the leaves on the left-hand side increase from left to right.

False. Leaves always increase as you move away from the stems. On the left-hand side that means they get larger from right to left, which is the mirror image of the right-hand side.

Why is a rough version of a stem-and-leaf diagram drawn first, and when can you skip it?

Because the data usually arrives out of order, so a rough version gets every value into the right stem before anything is arranged neatly. You can skip it when the data has already been given to you in order.

What check tells you a completed two-way table has no mistakes in it?

The row totals and the column totals must both add up to the same overall figure in the corner. That is because the corner figure is the whole data set counted twice over, once across the rows and once down the columns, so a disagreement means something is wrong.

Define two-way table .

A two-way table compares two different characteristics of the same group, with one set of categories across the columns and the other down the rows. Each cell gives the number having one particular combination of the two, such as Year 12 students who study Spanish.

On a Venn diagram of two sets, what do the numbers outside both circles but inside the rectangle represent?

The items that are in neither set. The rectangle stands for the entire data set, so everything is somewhere inside it, and whatever sits outside both circles has neither characteristic.

True or False? The numbers written in the regions of a Venn diagram are always frequencies.

False. The regions can hold percentages instead, in which case every number in the diagram has to add up to 100. Whatever they hold, the numbers across the whole diagram must account for the complete data set.

Define population pyramid .

A population pyramid is a diagram showing how a population is spread across age groups , with one horizontal bar for each group. It is usually split down the middle by gender, with females on one side and males on the other.

What can the horizontal scale of a population pyramid show?

Either the actual number of people in each age group, or, more often, a percentage. Where it is a percentage it may be of the total population or of that gender's population alone, so the labelling has to be read carefully before anything is calculated.

True or False? An age group showing 0.0% on a population pyramid must contain nobody at all.

False. When percentages are rounded to one decimal place, any group holding less than 0.05% of the population is shown as 0.0%. There may still be people in it.

On a population pyramid, how can you tell at a glance whether more males or more females are aged 40 and over?

Compare the two sides bar by bar, from the 40 to 49 group upwards. If every male bar in that range is longer than the female bar beside it, there are more males aged 40 and over, with no need to add anything up.

Define choropleth map .

A choropleth map shows data for the different regions of a geographical area by shading or patterning each region according to its value. Regions with similar values are given the same shade, so patterns across the whole area stand out at a glance.

How is a choropleth map shaded when colour is not available?

With different depths of grey, running from black for the densest values through to white for the least dense. Patterns such as cross-hatching can be used in the same way.

True or False? A choropleth map must be an accurate geographical map of the area it represents.

False. An idealised or abstract diagram can be used instead, such as a field divided into a grid of equal squares. What matters is that each region carries a shade matching its own value.

Why does a choropleth map need a key?

Because a shade or a pattern means nothing on its own. The key is what turns each shade into the range of values it stands for, so without one no region can be given a number.

A field at a concert is divided into 30 equal squares, and on the choropleth map the darkest squares all lie along one edge. What does that suggest?

That the crowd was densest along that edge, since the darkest shading stands for the largest numbers of people. The stage was most likely on that side of the field, because that is where people gather.

Define cumulative frequency .

Cumulative frequency is a running total of the frequencies, added up as you work down the table. Each row's figure is the number of data values up to and including the end of that row's class.

When is a cumulative frequency step polygon used instead of a cumulative frequency diagram?

When the data is discrete , such as the number of eggs in a nest. A cumulative frequency diagram is used for grouped continuous data instead.

Why does a cumulative frequency step polygon rise in vertical jumps rather than along a smooth curve?

Because the data is discrete, so nothing at all happens between one possible value and the next. The running total stays flat across the gap and then jumps at each value that actually occurs.

At which point of each class interval is the cumulative frequency plotted on a cumulative frequency diagram, and why?

At the upper bound , the end of the class interval. The values in a class could be anywhere within it, so they cannot all be guaranteed to have been counted until you reach the top of that class.

What shape does a cumulative frequency diagram always have?

A stretched S-shape that only ever rises. Because every figure on it is a running total, the curve can level off but can never come back down towards the horizontal axis.

Why are the median and quartiles read from a cumulative frequency diagram only estimates?

Because the diagram is built from grouped data, so the original values are unknown. Joining the points assumes the data is spread evenly across each class, which will not be exactly true.

Which five values do you need in order to draw a box plot?

The five values are: the lowest value that is not an outlier the lower quartile the median the upper quartile the highest value that is not an outlier

True or False? The box on a box plot can be drawn to any depth.

True. The depth of the box carries no information at all and can be drawn to suit the page. Everything the box tells you is in where it sits along the scale and how wide it is, which is the interquartile range.

True or False? The median line always sits in the middle of the box on a box plot.

False. The median line is drawn at the median value on the scale, which is usually not halfway between the two quartiles. Where it sits tells you whether the data bunches towards the lower quartile or the upper one.

How is an outlier shown on a box plot, and what happens to the whisker?

An outlier is marked with a cross beyond the end of a whisker. The whisker then stops at the last value that is not an outlier, rather than running all the way out to the outlier itself.

What two features of two data sets do box plots let you compare?

An average , using the medians, and a spread , using the interquartile ranges. A smaller interquartile range means the values are more tightly bunched, so that data set is the more consistent of the two.

Why are box plots useful for data containing a few extreme values, such as the prices of cars?

Because they split the data at the quartiles, so the bulk of it still shows clearly even when a few values sit far out. One sports car among forty-nine family cars appears as an outlier rather than distorting the whole picture.

Apart from plotting at midpoints, what two rules apply when joining up a frequency polygon?

Do not join the polygon down to the horizontal axis, unless one of the frequencies really is zero. Do not join the last point back to the first, which is why the result is an open shape rather than a true polygon.

Why is frequency density used instead of frequency on a histogram?

Because the class intervals usually have unequal widths, so plotting raw frequency would make wide classes look far more important than they are. Dividing by the class width puts every class on the same footing, measuring how densely the data is packed inside its interval.

True or False? The bars of a histogram touch each other, with no gaps between them.

True. A histogram shows continuous data grouped into class intervals, and consecutive intervals meet, so the bars touch. A gap would mean a range of values that no class covers at all.

When can a histogram's vertical axis be labelled 'frequency' rather than 'frequency density'?

When all the class intervals have equal widths. The bars are then all the same width, so their heights are already in proportion to the frequencies and can be read off directly.

How do you estimate how many values lie above a point partway across one bar of a histogram?

Find the area of just the part of the bar you want, by multiplying its frequency density by the width of that part only. This assumes the values are spread evenly across the whole class, so the answer can only be an estimate.

What has to match before two histograms can be compared?

They must use the same class intervals and the same frequency density scale. Otherwise bars of the same size on the two diagrams would stand for different numbers of values.

A frequency polygon reaches its highest point at 195 seconds. What does that tell you?

That the class containing 195 seconds is the modal class , since the polygon peaks where the frequency is greatest. It is acceptable to give 195 seconds itself as an estimate of the modal value.

What three things should guide your choice of a data representation?

The three considerations are: the target audience , and how familiar they are with statistics the nature of the data , since some diagrams suit discrete data and others continuous or bivariate data the particular strengths and weaknesses of each representation

Which representations suit an audience unfamiliar with statistics, and which suit experienced statisticians?

A simple visualisation such as a bar chart, pictogram or pie chart suits a general audience. A box plot, cumulative frequency diagram or histogram suits an audience used to statistics, because each needs some interpretation before it means anything.

GCSEStatisticsEdexcelHigherFlashcardsProcessing, Representing & Analysing DataTabulation, Diagrams & Representation

Tabulation, Diagrams & Representation (Edexcel GCSE Statistics: Higher): Flashcards

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FrontBar Charts, Line Graphs & Pictograms
Complete the sentence about bar charts.
On a bar chart the horizontal axis shows the different $______$ being counted, and the height of each bar shows its $______$ in the data.

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Cards in this collection (80)

Complete the sentence about bar charts.
On a bar chart the horizontal axis shows the different $______$ being counted, and the height of each bar shows its $______$ in the data.
The completed sentence is:
On a bar chart the horizontal axis shows the different outcomes being counted, and the height of each bar shows its frequency in the data.
Why are the bars on a bar chart drawn with gaps between them?
Because a bar chart shows discrete data, where each bar is a separate category rather than part of a continuous scale.
The gaps make it clear that there are no values in between.
True or False?
All the bars on a bar chart must be drawn with the same width.
True.
Bars on a bar chart are always drawn with equal widths, so that the height on its own represents the frequency.
If the widths varied, a taller bar would no longer reliably mean a larger frequency.
When is a vertical line graph a better choice than a bar chart?
When the data is numerical and there are a lot of different values to show, such as test scores out of 20.
Thin lines fit where wide bars would be crowded, so the spread of the data and any unusual values stand out clearly.
What is the difference between a dual bar chart and a bar-line chart?
A dual bar chart shows two data sets measuring the same variable, so the bars sit side by side on one shared scale.
A bar-line chart shows two different variables, one as bars and one as a line, so each can have its own scale: monthly rainfall as bars with monthly temperature as a line.
Define pictogram.
A pictogram represents frequencies using symbols instead of bars, with no axes at all.
A key states what one symbol is worth, and part-symbols stand for part-values, so if one symbol means 2 people then half a symbol means 1 person.
What is a composite bar chart, and what does the height of one coloured section show?
A composite bar chart stacks the categories within each bar, so the overall height of a bar gives the total frequency.
A coloured section shows the frequency for that category alone, read as the difference between the two ends of the section rather than as the value at its top.
Why can a composite bar chart tell you something a plain bar chart of the same totals cannot?
Because it shows the breakdown within each bar as well as the total.
A time slot can have the smallest total traffic of the day and still contain the largest number of lorries, and only the composite chart reveals that.
Complete the formula for the angle of a sector in a pie chart.
$angle = \frac{______}{total} \times______$
The completed formula is:
$angle = \frac{frequency}{total} \times 360$
Why does a pie chart show proportions rather than actual amounts?
Because the whole circle always represents the whole data set, whatever its size, so a sector's angle only tells you what share of the total it is.
Without being told the total frequency you cannot turn any sector back into a number of items.
A sector of a pie chart measuring 30 degrees represents 15 people. How many people does the whole pie chart represent?
The whole circle is 360 degrees, which is 12 times as much, so the total is $15 \times 12 = 180$ people.
Once you know how many people one degree stands for, any other sector can be worked out the same way.
True or False?
A pie chart labelled 'not drawn accurately' can still have its angles measured with a protractor.
False.
If a diagram is not drawn accurately then the angles on the page are not the true ones, so measuring them tells you nothing.
Work the angles out from the frequencies instead, or use an angle the diagram states.
What makes comparative pie charts different from two ordinary pie charts?
Their areas are drawn in the same ratio as the total frequencies of the two data sets, so a bigger circle really does mean more data.
Two ordinary pie charts drawn the same size make data sets of very different sizes look equal.
Complete the formula for the radius of the second of two comparative pie charts.
$r_{2} = r_{1} \times \sqrt{\frac{______}{F_{1}}}$
The completed formula is:
$r_{2} = r_{1} \times \sqrt{\frac{F_{2}}{F_{1}}}$
One comparative pie chart has a radius of 5 cm and represents 5000 people. A second has a radius of 3 cm. How many people does the second represent?
Scale the frequency by the ratio of the areas, which for circles means the ratio of $3^{2}$ to $5^{2}$ exactly.
$5000 \times \frac{9}{25} = 1800 people$
Why must a stem-and-leaf diagram always have a key?
Because the same diagram could stand for very different numbers.
A stem of 3 with a leaf of 6 could mean 36 or 3.6, and only the key says which.
Define stem-and-leaf diagram.
A stem-and-leaf diagram splits each value into a stem, its leading digits, and a leaf, its last digit, putting all the values that share a stem on one row.
It orders the data and groups it into classes while still showing every original value.
Complete the sentence about stem-and-leaf diagrams.
The stems act as the $______$ intervals, and the number of leaves on a stem gives the $______$ for that interval.
The completed sentence is:
The stems act as the class intervals, and the number of leaves on a stem gives the frequency for that interval.
In a stem-and-leaf diagram the median turns out to be a leaf of 1 on the stem of 2. What is the median?
The median is 21, not 1.
A leaf gives only one digit, so you must put the value back together with its stem before writing it down, which is a common place to lose marks.
How do you find the modal class from a stem-and-leaf diagram?
Look for the stem with the most leaves.
Each stem is a class interval, so the longest row of leaves is the class holding the most data.
What is a back-to-back stem-and-leaf diagram used for?
Comparing two related sets of data, such as boys against girls, on one diagram instead of two.
The two sets share a single central column of stems, with one set's leaves running out to the left and the other's to the right.
True or False?
On a back-to-back stem-and-leaf diagram, the leaves on the left-hand side increase from left to right.
False.
Leaves always increase as you move away from the stems.
On the left-hand side that means they get larger from right to left, which is the mirror image of the right-hand side.
Why is a rough version of a stem-and-leaf diagram drawn first, and when can you skip it?
Because the data usually arrives out of order, so a rough version gets every value into the right stem before anything is arranged neatly.
You can skip it when the data has already been given to you in order.
What check tells you a completed two-way table has no mistakes in it?
The row totals and the column totals must both add up to the same overall figure in the corner.
That is because the corner figure is the whole data set counted twice over, once across the rows and once down the columns, so a disagreement means something is wrong.
Define two-way table.
A two-way table compares two different characteristics of the same group, with one set of categories across the columns and the other down the rows.
Each cell gives the number having one particular combination of the two, such as Year 12 students who study Spanish.
In a two-way table, 15 children in total chose clay modelling and 13 of them were in class B. How many class A children chose clay modelling?
2, found by taking the class B figure away from the column total, $15 - 13 = 2$ children.
Every empty cell can be reached this way, by subtracting the known entries in a row or column from that row or column's total.
Complete the sentence about reading a Venn diagram.
The region where two circles overlap counts the items in set A $______$ set B, while the two circles taken together count the items in set A $______$ set B.
The completed sentence is:
The region where two circles overlap counts the items in set A and set B, while the two circles taken together count the items in set A or set B.
Note that 'A or B' includes everything in the overlap as well.
On a Venn diagram of two sets, what do the numbers outside both circles but inside the rectangle represent?
The items that are in neither set.
The rectangle stands for the entire data set, so everything is somewhere inside it, and whatever sits outside both circles has neither characteristic.
A Venn diagram shows 12 in A only, 4 in the overlap, 21 in B only and 8 outside both circles. How many items are in set B?
25, which is the 4 in the overlap plus the 21 in B only.
The whole of circle B counts as set B, including the part it shares with A, and the complete data set here is $12 + 4 + 21 + 8 = 45$ items.
True or False?
The numbers written in the regions of a Venn diagram are always frequencies.
False.
The regions can hold percentages instead, in which case every number in the diagram has to add up to 100.
Whatever they hold, the numbers across the whole diagram must account for the complete data set.
Define population pyramid.
A population pyramid is a diagram showing how a population is spread across age groups, with one horizontal bar for each group.
It is usually split down the middle by gender, with females on one side and males on the other.
What can the horizontal scale of a population pyramid show?
Either the actual number of people in each age group, or, more often, a percentage.
Where it is a percentage it may be of the total population or of that gender's population alone, so the labelling has to be read carefully before anything is calculated.
A population pyramid gives 5.4% of the population as female aged 0 to 9, and 7.4% as female aged 10 to 19. What percentage is female and under 20?
Add the two percentages together, giving $5.4 + 7.4 = 12.8$ percent of the population.
Age groups can be combined like this because every person falls into exactly one of them.
True or False?
An age group showing 0.0% on a population pyramid must contain nobody at all.
False.
When percentages are rounded to one decimal place, any group holding less than 0.05% of the population is shown as 0.0%.
There may still be people in it.
On a population pyramid, how can you tell at a glance whether more males or more females are aged 40 and over?
Compare the two sides bar by bar, from the 40 to 49 group upwards.
If every male bar in that range is longer than the female bar beside it, there are more males aged 40 and over, with no need to add anything up.
Define choropleth map.
A choropleth map shows data for the different regions of a geographical area by shading or patterning each region according to its value.
Regions with similar values are given the same shade, so patterns across the whole area stand out at a glance.
How is a choropleth map shaded when colour is not available?
With different depths of grey, running from black for the densest values through to white for the least dense.
Patterns such as cross-hatching can be used in the same way.
True or False?
A choropleth map must be an accurate geographical map of the area it represents.
False.
An idealised or abstract diagram can be used instead, such as a field divided into a grid of equal squares.
What matters is that each region carries a shade matching its own value.
Why does a choropleth map need a key?
Because a shade or a pattern means nothing on its own.
The key is what turns each shade into the range of values it stands for, so without one no region can be given a number.
A field at a concert is divided into 30 equal squares, and on the choropleth map the darkest squares all lie along one edge. What does that suggest?
That the crowd was densest along that edge, since the darkest shading stands for the largest numbers of people.
The stage was most likely on that side of the field, because that is where people gather.
Define cumulative frequency.
Cumulative frequency is a running total of the frequencies, added up as you work down the table.
Each row's figure is the number of data values up to and including the end of that row's class.
A cumulative frequency table gives 14 for $x < 20$ and 39 for $x < 40$ in a survey. What is the frequency of the class $20 \leq x < 40$ on its own?
25, found by subtracting one cumulative total from the next, $39 - 14 = 25$ values.
A cumulative total includes everything below it, so the difference between consecutive rows recovers the frequency of a single class.
When is a cumulative frequency step polygon used instead of a cumulative frequency diagram?
When the data is discrete, such as the number of eggs in a nest.
A cumulative frequency diagram is used for grouped continuous data instead.
Why does a cumulative frequency step polygon rise in vertical jumps rather than along a smooth curve?
Because the data is discrete, so nothing at all happens between one possible value and the next.
The running total stays flat across the gap and then jumps at each value that actually occurs.
At which point of each class interval is the cumulative frequency plotted on a cumulative frequency diagram, and why?
At the upper bound, the end of the class interval.
The values in a class could be anywhere within it, so they cannot all be guaranteed to have been counted until you reach the top of that class.
A cumulative frequency diagram is drawn for times starting from the class $25 \leq s < 30$ upwards. Which extra point has to be added, and where?
A point at the very start, placed at the lowest value in the table with a cumulative frequency of zero.
Here that is the point $(25, 0)$ on the graph, because nobody had finished before 25 seconds.
What shape does a cumulative frequency diagram always have?
A stretched S-shape that only ever rises.
Because every figure on it is a running total, the curve can level off but can never come back down towards the horizontal axis.
Complete the quartile positions used to read a cumulative frequency diagram holding $n$ values in total.
$lower quartile at \frac{n}{______}, upper quartile at \frac{______n}{4}$
The completed positions are:
$lower quartile at \frac{n}{4}, upper quartile at \frac{3 n}{4}$
True or False?
On a cumulative frequency diagram of 60 values, the median is found halfway between the 30th and 31st values.
False.
From a cumulative frequency diagram you read straight across from the value $\frac{n}{2}$ on the vertical axis, which is 30 here.
Halfway between the 30th and 31st is the method for a list of raw values, not for a diagram.
How do you find the 10th percentile from a cumulative frequency diagram?
Work out its position as $\frac{n p}{100}$ where $p$ is the percentile wanted, so with 60 values the 10th percentile sits at position 6.
Read across from 6 on the cumulative frequency axis to the curve, then straight down to the horizontal axis.
100 phone calls are shown on a cumulative frequency diagram, and reading up from 12 minutes gives 90. How many calls lasted longer than 12 minutes?
About 10: the 90 counts the calls up to 12 minutes, so subtract it from the total, $100 - 90 = 10$ calls.
A cumulative frequency reading always gives the number below a value, so any 'more than' question needs subtracting from the total.
Why are the median and quartiles read from a cumulative frequency diagram only estimates?
Because the diagram is built from grouped data, so the original values are unknown.
Joining the points assumes the data is spread evenly across each class, which will not be exactly true.
Which five values do you need in order to draw a box plot?
The five values are:
- the lowest value that is not an outlier
- the lower quartile
- the median
- the upper quartile
- the highest value that is not an outlier
Complete the sentence about the parts of a box plot.
The box covers the middle $______$ of the data, and each whisker covers a further $______$ of it.
The completed sentence is:
The box covers the middle 50% of the data, and each whisker covers a further 25% of it.
The box runs from the lower quartile across to the upper quartile.
True or False?
The box on a box plot can be drawn to any depth.
True.
The depth of the box carries no information at all and can be drawn to suit the page.
Everything the box tells you is in where it sits along the scale and how wide it is, which is the interquartile range.
True or False?
The median line always sits in the middle of the box on a box plot.
False.
The median line is drawn at the median value on the scale, which is usually not halfway between the two quartiles.
Where it sits tells you whether the data bunches towards the lower quartile or the upper one.
How is an outlier shown on a box plot, and what happens to the whisker?
An outlier is marked with a cross beyond the end of a whisker.
The whisker then stops at the last value that is not an outlier, rather than running all the way out to the outlier itself.
What two features of two data sets do box plots let you compare?
An average, using the medians, and a spread, using the interquartile ranges.
A smaller interquartile range means the values are more tightly bunched, so that data set is the more consistent of the two.
Why are box plots useful for data containing a few extreme values, such as the prices of cars?
Because they split the data at the quartiles, so the bulk of it still shows clearly even when a few values sit far out.
One sports car among forty-nine family cars appears as an outlier rather than distorting the whole picture.
A box plot has a lower quartile of 2 and an upper quartile of 7.5. What is the interquartile range?
The interquartile range is $7.5 - 2 = 5.5$ on the scale, which is exactly the width of the box.
It measures the spread of the middle half of the data.
At which point of each class interval is a frequency polygon plotted?
At the midpoint of the class interval.
For a class of $150 \leq t < 180$ the point goes at 165 on the horizontal axis, plotted against that class's frequency.
Apart from plotting at midpoints, what two rules apply when joining up a frequency polygon?
Do not join the polygon down to the horizontal axis, unless one of the frequencies really is zero.
Do not join the last point back to the first, which is why the result is an open shape rather than a true polygon.
Complete the formula for frequency density.
$frequency density = \frac{______}{class width}$
The completed formula is:
$frequency density = \frac{frequency}{class width}$
Why is frequency density used instead of frequency on a histogram?
Because the class intervals usually have unequal widths, so plotting raw frequency would make wide classes look far more important than they are.
Dividing by the class width puts every class on the same footing, measuring how densely the data is packed inside its interval.
One class holds 10 values across a width of 20, another holds 20 values across a width of 100. Which holds its data more densely?
The first: its frequency density is $10 \div 20 = 0.5$ compared with $20 \div 100 = 0.2$ for the second.
The second class has twice as many values but spreads them over five times the width, so its data is the more thinly spread.
A histogram bar covers $15 \leq w < 30$ and has a height of 0.6. How many data values does it represent?
9: the area of the bar gives the frequency, so multiply the height by the width, $0.6 \times 15 = 9$ values.
It is always the area of a bar and never its height that gives the frequency, so the tallest bar need not hold the most values.
True or False?
The bars of a histogram touch each other, with no gaps between them.
True.
A histogram shows continuous data grouped into class intervals, and consecutive intervals meet, so the bars touch.
A gap would mean a range of values that no class covers at all.
When can a histogram's vertical axis be labelled 'frequency' rather than 'frequency density'?
When all the class intervals have equal widths.
The bars are then all the same width, so their heights are already in proportion to the frequencies and can be read off directly.
How do you estimate how many values lie above a point partway across one bar of a histogram?
Find the area of just the part of the bar you want, by multiplying its frequency density by the width of that part only.
This assumes the values are spread evenly across the whole class, so the answer can only be an estimate.
What has to match before two histograms can be compared?
They must use the same class intervals and the same frequency density scale.
Otherwise bars of the same size on the two diagrams would stand for different numbers of values.
A frequency polygon reaches its highest point at 195 seconds. What does that tell you?
That the class containing 195 seconds is the modal class, since the polygon peaks where the frequency is greatest.
It is acceptable to give 195 seconds itself as an estimate of the modal value.
What three things should guide your choice of a data representation?
The three considerations are:
- the target audience, and how familiar they are with statistics
- the nature of the data, since some diagrams suit discrete data and others continuous or bivariate data
- the particular strengths and weaknesses of each representation
Which representations suit an audience unfamiliar with statistics, and which suit experienced statisticians?
A simple visualisation such as a bar chart, pictogram or pie chart suits a general audience.
A box plot, cumulative frequency diagram or histogram suits an audience used to statistics, because each needs some interpretation before it means anything.
A table and a bar chart present the same data. What does each one do better than the other?
The table gives the exact values, which a chart usually cannot.
The bar chart shows patterns and trends at a glance, which a table hides among its figures.
Mary wants the proportion of students preferring each topping. One school's data is a pie chart, the other's a bar chart. Which helps her more?
The pie chart, because it shows proportions directly, often labelled as percentages on the diagram itself.
The bar chart gives counts, so she would have to total them first and then work out each proportion for herself.
True or False?
A graph whose vertical scale does not start at zero has been drawn incorrectly.
False.
A scale does not have to start at zero, so the graph is not wrong.
It can still mislead badly: a bar drawn nearly five times as tall as another may represent an increase of only about 40%.
Why does an unequal scale distort a diagram?
Because equal distances along the axis no longer stand for equal amounts.
A bar twice as tall as another may then represent far less than twice the value, so the shapes plotted against the scale mislead the eye.
Complete the sentence about incomplete diagrams.
If a graph has no $______$ on its axes there is no way to know what the data shows, and if a chart has no $______$ it may be impossible to interpret at all.
The completed sentence is:
If a graph has no labels on its axes there is no way to know what the data shows, and if a chart has no key it may be impossible to interpret at all.
Why can a 3D pie chart misrepresent its data?
Because tilting the circle distorts the angles, so the sectors no longer look proportional to what they represent.
Sectors at the front look larger than they are, and sectors at the back look smaller or are hidden behind them altogether.
How can a histogram itself be drawn so that it misrepresents its data?
By working out the frequency densities incorrectly, or by plotting the class widths wrongly along the horizontal axis.
Either mistake changes the areas of the bars, and on a histogram it is the area that represents the frequency.

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