Cumulative Frequency & Box Plots Flashcards

All 27 cards in this deck

What is cumulative frequency?

A running total of the frequencies — the number of values up to and including the upper end of each class.

In a cumulative frequency table, how should the class intervals be written?
e.g. original classes $0<t\le10$, $10<t\le20$, $20<t\le30$

Every interval starts at the lowest value: $0<t\le10$, $0<t\le20$, $0<t\le30$ (all 'less than or equal to' values).

What should the last cumulative frequency in a completed table equal?

The total frequency — the total number of items in the data set.

True or false? The cumulative frequency of a class is just the frequency of that class.

False. You add all the frequencies up to and including that class.

What are the steps to complete a cumulative frequency table from a grouped frequency table?
e.g. frequencies $5$, $8$, $12$

1. Copy the first frequency: $5$
2. Add the next frequency: $5+8=13$
3. Keep adding along the table: $13+12=25$ (last value = total)

What is a cumulative frequency graph used for?

Estimating the median, quartiles and interquartile range of grouped data, and the number of values above or below a given value.

On a cumulative frequency graph, at which value of the data variable is each cumulative frequency plotted?
e.g. class $20<t\le40$ with cumulative frequency $35$

At the upper class boundary, so plot $(40,\,35)$.

True or false? Cumulative frequency points are plotted at the midpoint of each class interval.

False. They are plotted at the upper end (upper class boundary) of each interval.

How should the plotted points on a cumulative frequency graph be joined?

With a smooth curve or with straight line segments (a cumulative frequency polygon) — either is accepted, and no part may have a negative gradient.

On a cumulative frequency graph, what goes on each axis?

The data variable (labelled, with units) on the horizontal axis; cumulative frequency on the vertical axis.

Where does a cumulative frequency graph start?
e.g. first class $0<w\le2$

At the lower boundary of the first class with cumulative frequency $0$, i.e. the point $(0,\,0)$.

What are the steps to draw a cumulative frequency graph from a grouped frequency table?
e.g. classes $0<w\le2$ (frequency 7) and $2<w\le4$ (frequency 21)

1. Work out the cumulative frequencies: $7$, $28$
2. Plot each one at the upper class boundary: $(2,7)$, $(4,28)$
3. Join the points with a smooth curve or line segments

How do you estimate the median from a cumulative frequency graph?
e.g. a graph for $60$ students

Read across from cumulative frequency $\frac{n}{2}=30$ to the curve, then down to the horizontal axis.

At which cumulative frequencies do you read off the lower and upper quartiles?
e.g. a graph for $60$ students

Lower quartile at $\frac{n}{4}=15$ and upper quartile at $\frac{3n}{4}=45$; read across to the curve then down.

How do you estimate the number of values greater than a given value from a cumulative frequency graph?
e.g. of $60$ students, how many are taller than $160$ cm

Read the cumulative frequency at that value and subtract it from the total: if the reading at $160$ is $48$, the answer is $60-48=12$.

How do you estimate the number of values between two given values from a cumulative frequency graph?
e.g. between $40$ g and $60$ g

Read the cumulative frequency at each value and subtract: if the readings are $20$ at $40$ g and $50$ at $60$ g, the answer is $50-20=30$.

How do you estimate the 90th percentile from a cumulative frequency graph?
e.g. a graph for $80$ items

Read across at $90\%$ of the total, $0.9\times80=72$, then down to the horizontal axis.

What must your answer include when asked whether a claim about a cumulative frequency graph or box plot is valid?

A decision (yes or no) and a reason quoting values read from the graph.
e.g. 'No, the median is $57$ g, not $50$ g.'

What five values does a box plot show?

The least value, lower quartile, median, upper quartile and greatest value.

On a box plot, which distances give the range and which give the interquartile range?

Range = distance between the whisker ends (greatest $-$ least); IQR = width of the box (upper quartile $-$ lower quartile).

True or false? The line drawn inside the box of a box plot shows the mean.

False. It shows the median.

What are the steps to draw a box plot from a five-figure summary?
e.g. least $8$, LQ $25$, median $34$, UQ $42$, greatest $74$

1. Draw the box from the LQ to the UQ: $25$ to $42$
2. Draw a line inside the box at the median: $34$
3. Draw whiskers out to the least and greatest values: $8$ and $74$

When drawing a box plot using values read from a cumulative frequency graph, where do the whisker ends come from?

From the least and greatest values given in the question — the cumulative frequency graph does not show them.

When comparing two distributions shown by box plots, what two things must you compare?

A measure of average (the medians) and a measure of spread (the IQR or the range), with at least one comparison in the context of the question.

True or false? The median and quartiles found from a cumulative frequency graph are only estimates.

True. The original data is grouped, so exact values are unknown and readings from the curve are estimates.

How do you use a cumulative frequency graph to estimate the value that a given percentage of the data lies below?

e.g. the weight below which the lightest 25% of 60 potatoes lie

1. Find that percentage of the total: 25% of 60 = 15
2. Read across from cumulative frequency 15 to the curve, then down to the weight axis
3. That weight is the estimate

How do you estimate what percentage of the data is below a given value on a cumulative frequency graph?

e.g. the percentage of 80 plants shorter than 90 cm

Read the cumulative frequency at that value, then divide by the total and multiply by 100.
e.g. reading 74 gives $\frac{74}{80}\times100=92.5\%$