Same data, two headlines: "soaring" and "flat". How?

Tally charts, frequency tables and pictograms

• A tally chart records data as you collect it, in groups of five strokes.
• A frequency table lists each category or value with its frequency (how many). A two-way table sorts by two features at once.
• A pictogram uses a symbol for a fixed number of items, shown in a key: if one car symbol = 4 cars, then 321​31/2 symbols = 14 cars.
• Part-symbols must be easy to read: halves or quarters.

Bar charts: single, dual and composite

• A bar chart shows categorical or discrete data: equal widths, gaps between bars, and an even frequency scale from 0.
• A dual bar chart puts bars for two groups side by side, with a key.
• A composite (stacked) bar chart stacks the groups in one bar: its total height is the total frequency.
• Check what each small square is worth first.

How graphs can mislead

• Read the vertical axis before the picture. If it does not start at 0 (and has no break mark), small differences look huge.
• In chart B the April bar looks 7 times as tall as January's, yet April's 1,030 members are only 3% more than January's 1,000.
• Other tricks: unequal steps on a scale, missing labels, pictogram symbols of different sizes, tilted 3-D pie charts.

A firm's average pay is £60,000, yet 9 in 10 staff earn £25,000. How?

Mean, median, mode and range

• Mean: add all the values and divide by how many there are.
• Median: put the values in order; the median is the middle one. With an even number of values it is halfway between the middle two. Its position is the 2n+1​(n+1)/2th value.
• Mode: the value that occurs most often (there can be more than one, or none).
• Range $=$= largest value $-$− smallest value. It measures spread: how varied the data are.

Averages from a frequency table

• In a frequency table each value $x$x occurs $f$f times, so each row totals $fx$fx.

• Mode: the value with the highest frequency. Median: find its position, 2n+1​(n+1)/2, then use running totals.
• Pets owned by 25 people (0 pets: 7, 1: 9, 2: 6, 3: 2, 4: 1): the median is the 13th person, and running totals 7, 16, … show they own 1 pet. The mode is also 1.
• Divide by $\sum f$Σ f, not by the number of rows.

Estimated mean from grouped data

• In grouped data you do not know the exact values, so use the midpoint of each class to stand for every value in it.
• Estimated mean =∑f∑(midpoint×f)​= (Σ (midpoint × f))/(Σ f). It is an estimate because the values are not all at the midpoint.
• The modal class is the class with the highest frequency.
• The range can only be estimated: largest possible value minus smallest possible value.

Choosing an average and comparing data

• Mean: uses every value, but an outlier (far from the rest) can drag it a long way.
• Median: not affected by a few extreme values; best for data like pay or house prices.
• Mode: the only average for categorical data, such as favourite colour.
• To compare two data sets, compare an average and the range, in context: "P spent less time on average, and their times were more consistent."

Two classes have the same median mark. Were their results the same?

Quartiles and the interquartile range (Higher)

• In ordered data the median splits it in half; the lower and upper quartiles (LQ, UQ) split it into quarters.
• For $n$n listed values: LQ is the 4n+1​(n+1)/4th value, median the 2n+1​(n+1)/2th, UQ the 43(n+1)​(3(n+1))/4th.
• Interquartile range (IQR) $=$= UQ $-$− LQ: the spread of the middle half, unaffected by outliers.
• For 3, 4, 5, 7, 8, 9, 12: LQ is the 2nd value (4), median the 4th (7), UQ the 6th (9): IQR $=5$= 5.

Cumulative frequency graphs (Higher)

• Cumulative frequency is a running total: how many values are up to each upper class boundary.
• Plot each total at the upper class boundary, start with 0 at the lowest boundary, and join with a smooth curve.
• For $n$n values, read across from 2n​n/2 (median), 4n​n/4 (LQ) and 43n​3n/4 (UQ), then down.
• For how many are more than a value, read up from it and subtract from $n$n.

Comparing distributions (Higher)

• Two data sets can share a median yet be very different. Compare an average (the median) and a spread (the IQR).
• A smaller IQR means the middle half of the data is more consistent (closer together).
• Write each comparison in context: "The median time for Group A was lower, so on average Group A was quicker. Group A's IQR was smaller, so their times were more consistent."
• Use like for like: median with IQR, or mean with range.

Histograms and frequency density (Higher)

• A histogram shows continuous data with no gaps between bars. When class widths are unequal, the vertical axis is frequency density.

• The area of each bar represents the frequency: $\text{frequency}=\text{frequency density}\times \text{class width}$frequency = frequency density × class width.
• Example: the class $20<h\le 30$20 < h ≤ 30 has frequency 15 and width 10, so its bar has height $15÷10=1.5$15 ÷ 10 = 1.5.

A coin lands heads 7 times in a row. Is tails now "due"?

The probability scale

• Probability measures how likely an event is, from 0 (impossible) to 1 (certain). Evens is 0.5.
• Write probabilities as fractions, decimals or percentages, never as "1 in 4" or "1 : 4" in an answer.
• Words along the scale: impossible, unlikely, evens, likely, certain.
• A probability can never be below 0 or above 1. If you get 1.2, something has gone wrong.

Equally likely outcomes

• When outcomes are equally likely: P(event)=total number of outcomesnumber of outcomes in the event​P(event) = (number of outcomes in the event)/(total number of outcomes)
• A fair dice: P(5 or 6)=62​=31​P(5 or 6) = 2/6 = 1/3.
• $P(\text{not }A)=1-P(A)$P(not A) = 1 − P(A). A fair dice: P(not 6)=1−61​=65​P(not 6) = 1 − 1/6 = 5/6.
• The probabilities of all the possible outcomes, when they cannot happen together (mutually exclusive), add up to 1.
• "Not A" is written A′A', so P(A′)=1−P(A)P(A') = 1 − P(A).

Relative frequency and expected outcomes

• From an experiment: relative frequency =number of trialsnumber of times the event happened​= (number of times the event happened)/(number of trials). Use it as an estimate of the probability.
• The more trials, the closer relative frequency tends to get to the true probability. A few trials can wobble a lot. Each flip or spin is unaffected by earlier ones: a coin has no memory.
• Expected number of outcomes $=\text{probability}\times \text{number of trials}$= probability × number of trials. It is an estimate, not a promise.
• A fair (unbiased) coin, dice or spinner has equally likely outcomes. A large experiment whose relative frequencies are far from these suggests bias.

Roll two dice and add them. Which total is most likely, and why?

Sample spaces

• A sample space is the list of every possible outcome. List it systematically so you miss none and repeat none.
• Two coins: HH, HT, TH, TT. HT and TH are different outcomes (first coin, second coin).
• For two dice use a sample space diagram: a 6 by 6 grid with one cell for each of the 36 equally likely outcomes.
• When outcomes are equally likely:

Venn diagrams

• A Venn diagram sorts items into overlapping circles inside a rectangle. The overlap is in both sets; outside the circles is in neither.
• Fill in the overlap first, then the "only" parts, then outside. All regions sum to the total.
• Probability = (number in your region) ÷ (total in the rectangle).
• The universal set (rectangle) is $\xi$xi or $\mathcal{E}$ℰ; $A\cap B$A ∩ B is the overlap; $A\cup B$A ∪ B is A or B or both; A′A' is not A.

Roll a dice four times. Is at least one six better than evens?

The AND rule and the OR rule

• Events are independent if one happening does not change the probability of the other (two dice, a coin and a spinner, repeated spins).
• AND for independent events: multiply. $\text{P}(A\text{ and }B)=\text{P}(A)\times \text{P}(B)$P(A and B) = P(A) × P(B)
• OR for mutually exclusive events (they cannot both happen): add. $\text{P}(A\text{ or }B)=\text{P}(A)+\text{P}(B)$P(A or B) = P(A) + P(B)
• $\text{P}(\text{not }A)=1-\text{P}(A)$P(not A) = 1 − P(A), so P(at least one) $=1-$= 1 − P(none).
• Higher: the AND rule and tree diagrams are Higher tier; adding mutually exclusive probabilities is Foundation.

Tree diagrams (Higher)

• A tree diagram shows each stage as a set of branches. Write the outcome at the end of each branch and its probability beside it.
• The branches from any one point add to 1.
• Multiply along a path to get the probability of that combined outcome.
• Add the paths that give the event you want (the paths are mutually exclusive).
• Check: all the end probabilities together add to 1.

Without replacement (Higher)

• With replacement, the item goes back: the second-stage probabilities are the same as the first.
• Without replacement, the events are dependent: the second branches change because there is one fewer item.
• Take one away from the total on every second branch, and from the colour already taken on its own branch.
• Then multiply along and add the paths as before.

1 in 100 are ill; a 90% accurate test says you are. Chance you're ill?

"Given that" shrinks the total (Higher)

• A conditional probability is a probability given that something has already happened.
• "Given B" means you only look at the outcomes in B. B's total becomes the new denominator.

Conditional probability from a Venn diagram (Higher)

• "Given B": cover everything outside circle B. Circle B's total is the new denominator.
• The numerator is the overlap (the part of B that is also in A).
• If you are given set sizes and "neither", find the overlap first: overlap = A + B − (total − neither).

Reversing a tree with expected frequencies (Higher)

• The second branches of a tree are already conditional: "late, given it rained".
• To reverse the condition ("rained, given late"), imagine a round number, such as 100 or 1,000, going through the tree. These are expected frequencies.
• Add up every path that ends in the given outcome: that is the new total.
• Divide the path you want by that total.

Big ideas

• Choose the chart for the data: pictogram, bar chart (dual or composite) or pie chart for categories; read every value against the scale and key.
• Pie chart angle = frequency ÷ total × 360°; reverse it to find a frequency.
• A vertical axis that does not start at 0 can make a tiny change look dramatic.
• Mean =number of valuestotal​= total/(number of values); median = middle value of the ordered list; mode = most common; range = largest $-$− smallest.
• From a table, mean =∑f∑fx​= (Σ fx)/(Σ f); for grouped data use midpoints, and the answer is an estimate.
• Outliers drag the mean, not the median: choose the average that fits the data, and compare an average and a spread in context.
• Quartiles split ordered data into quarters; IQR = UQ − LQ measures the spread of the middle half.
• Cumulative frequency: plot running totals at upper class boundaries; read across from 2n​n/2, 4n​n/4, 43n​3n/4.
• Histograms: frequency density = frequency ÷ class width, and bar area gives frequency.
• Probability runs from 0 (impossible) to 1 (certain); $P(\text{not }A)=1-P(A)$P(not A) = 1 − P(A); all mutually exclusive outcomes sum to 1.
• Relative frequency estimates probability, and gets more reliable with more trials. Expected number = probability × trials.
• A fair coin has no memory: past results never make an outcome "due".
• List outcomes systematically; for two events use a grid. P(event) = favourable ÷ total, only when the outcomes are equally likely.
• In a two-way table or Venn diagram, find the missing values from the totals first, then divide by the total you are choosing from.
• $A\cap B$A ∩ B is the overlap, $A\cup B$A ∪ B is either or both, and A′A' is everything outside $A$A.
• AND (independent events): multiply. OR (mutually exclusive events): add. P(at least one) $=1-$= 1 − P(none).
• On a tree diagram, multiply along the branches and add the paths you want; each set of branches adds to 1.
• Without replacement, the second-stage probabilities change: one fewer in the total and in the group already taken.
• "Given B" means choose only from B: B's total is the denominator.
• From a two-way table or Venn diagram, divide the overlap by the total of the given group; to reverse a tree, use expected frequencies of a round number.