This note covers how data is collected and summarised: sampling methods and their weaknesses, types of data, measures of centre and spread, grouped data, outliers, and the main statistical diagrams. These ideas are the starting point for every later topic in statistics.
Key ideas
Populations and samples. A population is the whole group you want to know about. A sample is the part of it that you actually measure. A good sample is representative and has no bias, meaning no built-in tendency to favour some members of the population over others.
- Simple random sampling. Every member (and every group of the chosen size) has an equal chance of selection. Random numbers from a GDC or a list can be used.
- Systematic sampling. Pick a random start and then take every \( k \)-th member from an ordered list.
- Stratified sampling. Split the population into groups (strata) such as year groups, then take a random sample from each, in proportion to the size of the group: \( \text{sample from stratum} = \dfrac{\text{stratum size}}{\text{population size}} \times \text{sample size} \).
- Quota sampling. Similar to stratified, but the interviewer chooses people to fill fixed quotas, so the choice inside each group is not random.
- Convenience sampling. Use whoever is easiest to reach. It is quick but often biased.
Other sources of bias include a poor sampling frame (a list that leaves out part of the population), leading questions and non-response. A larger sample is generally more reliable, but size does not fix bias.
Types of data. Discrete data takes separate values, usually counts (number of siblings). Continuous data can take any value in a range, usually measurements (time, mass). Data placed into categories is categorical.
Measures of centre. The mean \( \bar{x} = \dfrac{\sum x}{n} \) uses every value and is affected by extreme values. The median is the middle value when the data is in order (the mean of the two middle values when \( n \) is even). The mode is the most frequent value. For grouped data, we speak of the modal class.
Measures of spread. The range is the largest minus the smallest value. The quartiles \( Q_1 \) and \( Q_3 \) split ordered data into quarters, and the interquartile range is \( \text{IQR} = Q_3 - Q_1 \). The standard deviation \( \sigma \) measures typical distance from the mean; the variance is \( \sigma^2 \). Use the one-variable statistics function on your GDC. Standard deviation and IQR describe how spread out the data is, and neither can be negative.
Outliers. A value is treated as an outlier if it is more than \( 1.5\times\text{IQR} \) above \( Q_3 \) or below \( Q_1 \). On a box plot, outliers are shown as separate crosses, and the whiskers go to the most extreme values that are not outliers.
Grouped data. When data is in classes, use the midpoint of each class as the value. The mean and standard deviation found this way are estimates, since the original values are unknown. Enter midpoints and frequencies as two lists on your GDC.
Diagrams. A histogram has continuous data on the horizontal axis and bars touching each other (for equal class widths, the height shows frequency). A cumulative frequency graph plots running totals against the upper class boundaries; read the median and quartiles from it at \( \tfrac12 n \), \( \tfrac14 n \) and \( \tfrac34 n \). A box-and-whisker plot shows the minimum, \( Q_1 \), median, \( Q_3 \) and maximum.
Effect of changing every value. If you add a constant \( c \) to every value, the mean, median and mode increase by \( c \), while the standard deviation, IQR and range do not change. If you multiply every value by \( k \), the centre measures are multiplied by \( k \), and the spread measures are multiplied by \( |k| \).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \( \bar{x} = \dfrac{\sum x}{n} \) | Mean of raw data. |
| \( \bar{x} = \dfrac{\sum fx}{n} \) | Mean from a frequency table, where \( n = \sum f \). For grouped data \( x \) is the class midpoint, and the result is an estimate. |
| \( \text{IQR} = Q_3 - Q_1 \) | Spread of the middle half of the data. |
| Outlier rule | Values below \( Q_1 - 1.5\,\text{IQR} \) or above \( Q_3 + 1.5\,\text{IQR} \). |
| \( \sigma \), \( \sigma^2 \) | Standard deviation and variance, from the GDC. |
| \( \text{sample size for a stratum} \) | \( \dfrac{\text{stratum size}}{\text{population}} \times \text{total sample} \). |
| \( \text{new mean} = k\bar{x} + c \) | If every value becomes \( kx + c \). The new standard deviation is \( |k|\sigma \). |
Worked examples
Example 1 A school has 500 students in Year 9, 400 in Year 10 and 300 in Year 11. A survey needs a stratified sample of 60 students. How many come from each year?
The population is \( 500 + 400 + 300 = 1200 \).
Year 9: \( \dfrac{500}{1200} \times 60 = 25 \). Year 10: \( \dfrac{400}{1200} \times 60 = 20 \). Year 11: \( \dfrac{300}{1200} \times 60 = 15 \).
Check: \( 25 + 20 + 15 = 60 \).
Answer: 25 from Year 9, 20 from Year 10 and 15 from Year 11, each chosen at random within the year.
Example 2 Twelve patients waited the following times (minutes) at a clinic: 8, 11, 12, 14, 15, 17, 18, 21, 22, 24, 27, 45. Find the mean, median, standard deviation and IQR, and decide whether 45 is an outlier.
Mean: \( \sum x = 234 \), so \( \bar{x} = \dfrac{234}{12} = 19.5 \).
Median: \( n = 12 \), so the median is halfway between the 6th and 7th values: \( \dfrac{17 + 18}{2} = 17.5 \).
Standard deviation: enter the data in the one-variable statistics function on your GDC and read the population standard deviation \( \sigma \): \( \sigma \approx 9.38 \).
Quartiles: the lower half is 8, 11, 12, 14, 15, 17, so \( Q_1 = \dfrac{12 + 14}{2} = 13 \). The upper half is 18, 21, 22, 24, 27, 45, so \( Q_3 = \dfrac{22 + 24}{2} = 23 \). Then \( \text{IQR} = 23 - 13 = 10 \).
Upper limit \( = 23 + 1.5 \times 10 = 38 \). Since \( 45 \gt 38 \), the value 45 is an outlier. (Lower limit \( = 13 - 15 = -2 \); no values are below it.)
Answer: mean 19.5 min, median 17.5 min, \( \sigma \approx 9.38 \) min, IQR 10 min; 45 is an outlier.
Example 3 The journey times of 50 cycle couriers are grouped: \( 0\le t\lt10 \): 4; \( 10\le t\lt20 \): 11; \( 20\le t\lt30 \): 19; \( 30\le t\lt40 \): 12; \( 40\le t\lt50 \): 4. Estimate the mean, state the modal class and estimate the median.
Midpoints: 5, 15, 25, 35, 45.
\( \sum fx = 4(5) + 11(15) + 19(25) + 12(35) + 4(45) = 20 + 165 + 475 + 420 + 180 = 1260 \).
Estimated mean \( = \dfrac{1260}{50} = 25.2 \) minutes.
The modal class is the class with the highest frequency: \( 20\le t\lt30 \).
Cumulative frequencies: 4, 15, 34, 46, 50. The median is the 25th value, which lies in the class \( 20\le t\lt30 \). Reading from a cumulative frequency graph (or by linear interpolation) gives \( 20 + \dfrac{25 - 15}{19} \times 10 \approx 25.3 \).
Answer: mean about 25.2 min; modal class \( 20\le t\lt30 \); median about 25.3 min.
Common errors
- Common error: calling a grouped-data mean "the mean". It is an estimate, because midpoints replace the real values.
- Common error: finding the median of unordered data. Sort the values first.
- Common error: thinking a bigger sample removes bias. If the method of choosing is biased, a larger sample stays biased.
- Common error: plotting a cumulative frequency graph against the class midpoints. Use the upper class boundaries.
- Common error: forgetting that adding a constant to all data does not change the spread.
Exam tip: When asked to comment on a sampling method, name the method, then give one specific reason it could be biased or unrepresentative for this population. A general statement such as "it is not fair" earns little.
Check yourself
- A researcher stands outside a gym and surveys the first 40 people to leave about their exercise habits for a town-wide study. Name the sampling method and give one weakness.
Answer
Convenience sampling. People at a gym exercise more than the typical resident, so the sample is biased towards frequent exercisers and does not represent the town. - A data set has mean 12 and standard deviation 3. Every value is replaced by \( 2x + 3 \). Find the new mean and standard deviation.
Answer
New mean \( = 2(12) + 3 = 27 \). New standard deviation \( = 2 \times 3 = 6 \) (the \( +3 \) does not affect spread). - For the data 3, 5, 5, 7, 8, 9, 10, 12, find the median, \( Q_1 \), \( Q_3 \) and IQR.
Answer
Median \( = \dfrac{7+8}{2} = 7.5 \). Lower half 3, 5, 5, 7 gives \( Q_1 = 5 \). Upper half 8, 9, 10, 12 gives \( Q_3 = \dfrac{9+10}{2} = 9.5 \). IQR \( = 9.5 - 5 = 4.5 \). - A population has 840 adults and 360 children. A stratified sample has 42 adults. How many children are sampled?
Answer
Sampling fraction \( = \dfrac{42}{840} = \dfrac{1}{20} \). Children \( = \dfrac{360}{20} = 18 \).