Course

Cambridge O Level Mathematics: Sets, Probability and Statistics (4024)

Name: Cambridge O Level Mathematics: Sets, Probability and Statistics (4024)
Price: 1499 BDT

Sets and Venn diagrams, then every statistics topic in section 9 from tally charts to histograms, then probability from a single event to tree diagrams without replacement. Each chapter ends with practice, exam questions and a summary.

100 lessons15h 34mEnglish

What you'll learn

Use set notation, shade a region from its notation and fill a Venn diagram with two or three sets from a word problem
Classify data and organise it in tally charts, grouped frequency tables and two-way tables
Draw and read bar charts, pictograms, pie charts and frequency diagrams
Calculate the mean, median, mode and range, from a list, a frequency table or grouped data, and choose the right average
Compare two sets of data with an average and a measure of spread, and say when a conclusion is not safe
Describe correlation on a scatter diagram and use a line of best fit to estimate
Read the median, quartiles, interquartile range and percentiles from a cumulative frequency diagram
Draw and read a histogram with unequal class widths using frequency density
Find probabilities from equally likely outcomes, from experiments and from tables or Venn diagrams
Use sample space diagrams and tree diagrams for combined events, with and without replacement

Requirements

Number at O Level standard, or our Number course: fractions, decimals, percentages and rounding
Solving a simple linear equation, for Venn diagram and missing-value questions
A ruler, a protractor and a pair of compasses for charts, and a calculator

Course content

100 lessons · 15h 34m

1.2 · Sets and Venn diagrams

The language of sets from zero: elements, n(A), set-builder notation, the universal set, complements, subsets and the empty set. Then Venn diagrams with two and three sets, shading and word problems.

0/13

Sets, elements and set notation

A set is a list with a clear rule. The symbol ∈ means "is an element of", and n(A) counts the elements.

Preview10m 2.

Describing a set in set-builder notation

Read {x : x is an even number} out loud as a sentence, then list the elements it allows.

Preview10m

The universal set and the complement

The universal set holds everything being talked about. A' is everything in it that is not in A.

10m

Subsets and the empty set

A ⊆ B when every element of A is also in B. The empty set ∅ has no elements at all.

10m

Venn diagrams

A box for the universal set and a circle for each set. Every element goes in exactly one region.

10m

Intersection of sets

A ∩ B holds the elements that are in A and also in B: the overlap of the two circles.

10m

Union of sets

A ∪ B holds everything in A or in B or in both, and each element is written only once.

10m

Shading regions of a Venn diagram

Turn notation like A' ∩ B into a region: work out the bracket first, then the outside.

12m

Solving problems with Venn diagrams

Put the number in the overlap first, then fill the rest. Use x for an unknown and make an equation.

12m

10.

Venn diagrams with three sets

Three circles make eight regions. Fill the centre first and work outwards.

12m

11.

Sets and Venn diagrams: practice

A timed drill of mixed questions on notation, regions and Venn diagram counts, each with a worked reason.

12.

Sets and Venn diagrams: exam practice

Past-paper style questions with full working and mark schemes: list, shade, fill and count.

15m

13.

Sets and Venn diagrams: summary

Every symbol, the shading method and the common mistakes on one screen.

9.1 · Classifying and tabulating data

Sort data into types, then put it in order: tally charts, frequency tables, grouped frequency tables with class intervals, and two-way tables.

0/8

Types of data: qualitative and quantitative

Data in words, like a favourite sport, is qualitative. Data in numbers, like a height, is quantitative.

Preview9m 2.

Discrete and continuous data

Discrete data is counted and jumps between values. Continuous data is measured and can take any value in a range.

Preview9m

Tally charts and frequency tables

Tally in groups of five, then count each row into a frequency. The frequencies must add to the total.

10m

Grouped frequency tables

Many different values go into classes. Classes must not overlap, and 150 ≤ h < 160 says exactly where 160 goes.

12m

Two-way tables

Rows for one question, columns for another. Row totals and column totals both add to the same grand total.

12m

Classifying and tabulating data: practice

A timed drill: name the type of data, finish a frequency table and fill the gaps in a two-way table.

10m

Classifying and tabulating data: exam practice

Past-paper style questions with full working and mark schemes, ending on a two-way table.

12m

Classifying and tabulating data: summary

The types of data, the rules for class intervals and the table checks on one screen.

9.4 · Statistical charts and diagrams

Draw and read the charts on every exam paper: bar charts, dual and composite bar charts, pictograms, pie charts and simple frequency distributions.

0/10

Bar charts

Equal widths, equal gaps, a labelled scale from zero. The height of each bar is its frequency.

Preview10m 2.

Dual (side-by-side) bar charts

Two bars side by side for each category, with a key, so two groups can be compared at a glance.

Preview10m

Composite (stacked) bar charts

The parts are stacked into one bar. Read each part as the top of its section minus the bottom.

10m

Pictograms

Read the key first. Half a symbol stands for half of what one whole symbol stands for.

Drawing pie charts

Each group gets its fraction of 360°. The angles must add to 360° before you draw anything.

12m

Interpreting pie charts

Go backwards: angle ÷ 360° is the fraction, and the fraction times the total is the frequency.

10m

Simple frequency distributions

Grouped continuous data drawn with no gaps between the bars, so the shape of the data shows.

10m

Statistical charts and diagrams: practice

A timed drill: read a bar, count a pictogram, find a pie-chart angle and choose the right chart.

Statistical charts and diagrams: exam practice

Past-paper style questions with full working and mark schemes: complete a chart and read one back.

15m

10.

Statistical charts and diagrams: summary

Which chart for which data, the pie-chart rule and the common drawing mistakes on one screen.

9.3 · Averages and measures of spread

The mean, median, mode and range, and which average to use. Then averages from a frequency table, an estimate of the mean for grouped data, and the modal class.

0/12

The mean

Add all the values and divide by how many there are. It is the fair share if the total were split equally.

Preview10m 2.

The median

Put the values in order and take the middle one. With an even number of values, use halfway between the middle two.

Preview10m

The mode and the range

The mode is the most common value. The range, largest minus smallest, measures how spread out the data is.

10m

Choosing the best average

One very large value pulls the mean but not the median. The mode is the only average for data in words.

10m

Using the mean to find a missing value

Mean × number of values gives the total. Take away the values you know to find the one you do not.

10m

The mean from a frequency table

Multiply each value by its frequency, add the products, then divide by the total frequency: Σfx ÷ Σf.

12m

The median and mode from a frequency table

The mode has the highest frequency. For the median, count down the running totals to the middle position.

10m

An estimate of the mean for grouped data

The exact values are lost in the classes, so use each midpoint. That is why the answer is only an estimate.

12m

The modal class

The class with the highest frequency. Also find the class that contains the median by counting positions.

10.

Averages and measures of spread: practice

A timed drill: the three averages and the range from lists, from frequency tables and from grouped data.

12m

11.

Averages and measures of spread: exam practice

Past-paper style questions with full working and mark schemes, including an estimated mean from a table.

15m

12.

Averages and measures of spread: summary

The formulas, when to use each average and the common mistakes on one screen.

9.2 · Interpreting statistical data

Read tables and charts, say what they show, compare two sets of data with an average and a measure of spread, and know when a graph or a small survey cannot support a conclusion.

0/9

Reading and interpreting tables and diagrams

Read the title, the key and the scale before any number. Then answer exactly what the question asks.

Preview10m 2.

Drawing inferences from data

An inference is a sensible statement the data supports. Back each one with a number from the table or chart.

Preview10m

Comparing sets of data using averages and range

Make two comparisons: one average for the typical value, one range for the spread, each with numbers and context.

12m

Comparing sets of data using tables and graphs

Put the two groups on the same scale. Then compare where the data sits and how widely it spreads.

10m

Misleading graphs

A scale that does not start at zero, uneven steps or bars of different widths can make a small change look huge.

10m

Restrictions on drawing conclusions

A small sample, or a sample of only one kind of person, cannot speak for everyone. Say what the data can and cannot show.

10m

Interpreting statistical data: practice

A timed drill: read a value, judge a statement, compare two groups and spot the misleading graph.

10m

Interpreting statistical data: exam practice

Past-paper style questions with mark schemes, including the two-mark "compare" answer examiners want.

15m

Interpreting statistical data: summary

How to compare, what makes a graph misleading and the limits of a survey on one screen.

9.5 · Scatter diagrams

Plot pairs of values as crosses, describe the correlation, draw a line of best fit by eye and use it to estimate, knowing when the estimate can be trusted.

0/8

Drawing scatter diagrams

Each pair of values becomes one cross. Choose a scale for each axis that fits all the data.

Preview10m 2.

Positive, negative and zero correlation

Both go up together: positive. One goes up as the other goes down: negative. No pattern: zero correlation.

Preview10m

Interpreting scatter diagrams

Describe the link in context, pick out a point that does not fit, and remember correlation does not prove cause.

10m

Drawing a line of best fit by eye

One ruled straight line through the middle of the crosses, with about the same number on each side.

10m

Using a line of best fit to make estimates

Go across to the line and down to read an estimate. Inside the data it is reliable; far outside it is not.

12m

Scatter diagrams: practice

A timed drill: name the correlation, judge a line of best fit and decide whether an estimate is safe.

Scatter diagrams: exam practice

Past-paper style questions with mark schemes: plot, describe, draw the line and estimate.

15m

Scatter diagrams: summary

The three kinds of correlation, the rules for a line of best fit and the common mistakes on one screen.

9.6 · Cumulative frequency diagrams

Build a cumulative frequency table, draw the curve, and read off the median, quartiles, interquartile range and percentiles. Then use the curve to answer questions and compare two groups.

0/9

Cumulative frequency tables

A running total: each row says how many values are less than or equal to its upper boundary.

Preview10m 2.

Drawing cumulative frequency diagrams

Plot each running total at the upper class boundary, mark it with a cross and join with a smooth curve.

Preview12m

Estimating the median from a cumulative frequency diagram

Go across from half the total frequency to the curve, then down to the value axis.

10m

Quartiles and the interquartile range

Read across at one quarter and three quarters of the total. The gap between them is the spread of the middle half.

12m

Percentiles

The 90th percentile is the value with 90% of the data below it. Find 90% of the total and read across.

10m

Interpreting cumulative frequency diagrams

Read up from a value to find how many are below it, subtract to find how many are above, and compare two groups.

12m

Cumulative frequency diagrams: practice

A timed drill: running totals, where to plot, and which position to read for each measure.

Cumulative frequency diagrams: exam practice

Past-paper style questions with mark schemes: complete the table, draw the curve, read the median and IQR.

15m

Cumulative frequency diagrams: summary

Where to plot, which positions to read and the common mistakes on one screen.

9.7 · Histograms

From equal classes to unequal ones: frequency density, drawing a histogram so the area of each bar is its frequency, and reading frequencies and estimates back from a histogram.

0/8

Histograms with equal class widths

Bars touch because the data is continuous. When every class is the same width, height shows frequency.

Preview10m 2.

Frequency density

Frequency density = frequency ÷ class width. It makes the area of each bar equal to its frequency.

Preview10m

Drawing histograms with unequal class widths

Add a frequency density column to the table, then draw each bar at that height across its full class width.

12m

Interpreting histograms

Go backwards: frequency = frequency density × class width. Use it to complete a table from a histogram.

12m

Estimating from a histogram

Take the part of a bar you need: its area gives an estimate of how many values lie in part of a class.

12m

Histograms: practice

A timed drill: class widths, frequency densities, frequencies from bars and the modal class.

10m

Histograms: exam practice

Past-paper style questions with mark schemes: complete a histogram and a table, then estimate.

15m

Histograms: summary

The frequency density rule, area as frequency and the common mistakes on one screen.

8.1-8.2 · Probability

The probability scale, P(A) notation, the probability of a single event and of it not happening. Then relative frequency from experiments, fair and biased, and expected frequency.

0/11

The probability scale

Every probability sits from 0 (impossible) to 1 (certain). Even chance is exactly halfway, at 0.5.

Preview9m 2.

Calculating the probability of a single event

When outcomes are equally likely, P(A) = number of ways A can happen ÷ total number of outcomes.

Preview10m

The probability of an event not occurring

Either A happens or it does not, so P(A') = 1 − P(A).

Probability from tables, graphs and Venn diagrams

Find the count you want and the total in the table or diagram. The probability is one over the other.

10m

Relative frequency

Relative frequency = number of times it happened ÷ number of trials. It comes from doing an experiment.

10m

Relative frequency as an estimate of probability

The more trials, the closer the relative frequency usually gets to the true probability.

10m

Fair, biased and random

Fair means every outcome is equally likely. A biased spinner shows up in many trials, not in a few.

10m

Expected frequency

Expected frequency = probability × number of trials. It is what you expect on average, not a promise.

10m

Probability: practice

A timed drill: place events on the scale, find P(A) and P(not A), relative and expected frequency.

10.

Probability: exam practice

Past-paper style questions with mark schemes: a probability table with a missing value, then expected frequency.

15m

11.

Probability: summary

The scale, the formulas, the words fair, biased and random, and the common mistakes on one screen.

10.

8.3 · Probability of combined events

Two or more events together: sample space diagrams, Venn diagrams, when to add and when to multiply, and tree diagrams with and without replacement, up to three stages and "at least one".

0/2

Sample space diagrams

List every possible pair of results in a grid or a list, so no outcome is missed or counted twice.

Preview10m 2.

Probability from sample space diagrams

Two dice give 36 equally likely cells. Circle the cells you want and divide by 36.

Preview10m

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.