Can asking 1,000 people tell you what a whole country thinks?

Populations and samples

• The population is the whole group you want to know about. A sample is the part of it you actually collect data from.
• Samples are quicker and cheaper; sometimes testing destroys the item (testing how long batteries last).
• You use the sample to infer (estimate) facts about the population.
• Limitations: a sample gives only an estimate, and a different sample would give slightly different results.

Random samples and bias

• In a simple random sample, every member of the population has an equal chance of being chosen.
• Method: number every member, then use a random number generator (or names from a hat); ignore repeats.
• A sample is biased if it is likely to misrepresent the population, e.g. asking people at a gym how often they exercise.
• Larger random samples give more reliable estimates. A larger biased sample is still biased.

Types of data

• Qualitative (categorical) data are words or categories: favourite sport, eye colour.
• Quantitative data are numbers. Discrete data are counted and take separate values (number of siblings). Continuous data are measured and can take any value in a range (height, time, mass).
• The type of data decides which charts and averages make sense.

Designing a questionnaire

• A good question is clear, not leading ("Don't you agree that…?") and not too personal.
• Response boxes must cover every possible answer and not overlap. Give a time frame ("per week").

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.
• A vertical line chart uses thin lines instead of bars, for ungrouped discrete data such as goals scored.
• Check what each small square is worth first.

Line graphs: time series and frequency polygons

• A time series graph plots a quantity against time (horizontal axis), with points joined by straight lines.
• Look for the trend (general rise or fall) and any seasonal pattern that repeats each year. Here sales peak every third quarter.
• A frequency polygon shows grouped data: plot each frequency at its class midpoint (5, 15, 25, … for 0–10, 10–20, …) and join with straight lines.
• Two polygons on the same axes are easy to compare.

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 median class is the class containing the 2n+1​(n+1)/2th value.
• 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."
• An outlier may be an error: check it before removing it.

Ice-cream sales and sunburn rise together. Does ice cream cause it?

Scatter graphs and correlation

• A scatter graph plots pairs of data (two variables for each item) as points, usually crosses. Each axis has its own scale.
• Positive correlation: as one variable increases, the other tends to increase.
• Negative correlation: as one increases, the other tends to decrease.
• No correlation: no clear pattern.
• Points close to a line show strong correlation; points loosely spread show weak correlation.

Lines of best fit

• A line of best fit is a single straight ruled line, drawn by eye, that follows the trend.
• It runs across the whole data set with roughly the same number of points on each side. It need not pass through any point or the origin.
• Ignore any outlier (a point that does not fit the pattern).
• To estimate, go from a known value to the line, then across to the other axis.

Interpolation, extrapolation and outliers

• Interpolation: estimating inside the range of the data. Usually reliable if the correlation is strong.
• Extrapolation: estimating outside the range of the data. Risky: the trend may not continue.
• An outlier may be an error or an unusual case (a day the shop shut early). Leave it out of the line.
• A small sample or weak correlation gives only rough estimates.

Correlation does not prove causation

• Correlation shows that two variables change together. It does not show that one causes the other.
• Often a third factor drives both. Example: over a summer, sales of electric fans and of cold drinks rise together; hot weather causes both.
• Sometimes the link is coincidence, especially with small data sets.
• To show cause you need more than a scatter graph, such as a controlled experiment. In an exam, say "correlation does not mean causation" and name a possible third factor.

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.
• With equal class widths the vertical axis can simply show frequency.

Big ideas

• A sample is used to estimate facts about a population; the estimate is only as good as the sample.
• Random samples avoid bias; bigger random samples are more reliable, but size never cures bias.
• Data are qualitative or quantitative; quantitative data are discrete (counted) or continuous (measured).
• 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.
• Time series show trends and seasonal patterns; frequency polygons plot frequencies at class midpoints.
• Stem-and-leaf diagrams keep every value: ordered leaves and a key.
• 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.
• Scatter graphs show positive, negative or no correlation.
• Describe the strength too: strong or weak.
• A line of best fit is a straight ruled line through the trend; use it to estimate, trusting interpolation more than extrapolation.
• Correlation does not mean causation: look for a third factor.
• 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.
• A box plot shows lowest, LQ, median, UQ, highest on a scale; compare box plots on the same scale.
• Histograms: frequency density = frequency ÷ class width, and bar area gives frequency.