DP Maths AI HL 4.2 Descriptive Statistics

DP Maths AI HL 4.2 Descriptive Statistics - Full notes

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DP Maths AI HL | 4.2 Descriptive Statistics

DP MATHEMATICS AI · TOPIC 4: STATISTICS AND PROBABILITY

4.2

Descriptive Statistics

Full notes for Applications and Interpretation HL

Contents

1	Populations, samples and types of data
2	Sampling techniques and bias
3	Reliability of data and outliers
4	Frequency tables and grouped data
5	Histograms and frequency polygons
6	Cumulative frequency graphs
7	Box-and-whisker plots
8	Measures of central tendency
9	Estimating the mean of grouped data
10	Measures of dispersion
11	Variance and standard deviation
12	Effect of constant changes
13	Comparing distributions
14	Bivariate data and scatter diagrams
15	Pearson's correlation coefficient
16	The regression line
17	Spearman's rank correlation
18	Mixed review: worked problems
19	Combining data sets and weighted means (HL)
20	Non-linear regression and the coefficient of determination (HL)
21	Investigation: is there a link between sleep and reaction time?
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
distinguish populations, samples, discrete and continuous data
describe sampling techniques and identify bias
construct and interpret histograms, cumulative frequency graphs and box plots
identify outliers using 1.5 × IQR
calculate and interpret measures of centre and spread with technology
compare distributions in context
find and interpret Pearson's r and the regression line
calculate and interpret Spearman's rank correlation

HL booklet: this booklet contains all the SL content plus combined data sets and non-linear regression with R2.

Guiding questions

How can data be collected fairly and summarized clearly? How do we describe and compare distributions and relationships?

How this topic is examined

Descriptive statistics appears in every AI examination. Paper 1 questions ask for summary statistics, outliers, or reading graphs; Paper 2 questions build on them: comparing two groups, interpreting correlation and regression, and evaluating sampling methods.

Always compare a measure of centre and a measure of spread, interpret in context, and state the technology used.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Populations, samples and types of data

What are we measuring?

Think about it

A school wants to know how much time its 1500 students spend on homework. Is it better to ask all of them, or a carefully chosen sample? What could go wrong with a sample?

The population is the whole group of interest; a sample is the part from which data are collected. Summary values of a sample (statistics) estimate those of the population (parameters).

Qualitative (categorical) data are non-numerical, such as eye colour. Quantitative data are numerical: discrete (counted, such as number of siblings) or continuous (measured, such as height or time).

Practice exercise 1

1. Classify: (a) shoe size; (b) blood type; (c) time to run 100 m.

2. Explain the difference between a parameter and a statistic.

2 Sampling techniques and bias

Choosing a representative sample.

Method	Description	Advantage / disadvantage
Simple random	every member equally likely to be chosen	unbiased; needs a full list
Systematic	every kth member after a random start	simple; biased if the list is patterned
Stratified	random samples from groups in proportion to size	represents subgroups; needs group sizes
Quota	fixed numbers from groups, not random	quick; can be biased
Convenience	easily available members	cheap; often biased

WORKED EXAMPLE 1 Stratified sample (standard)

A company has 240 office workers, 560 factory workers and 200 drivers. A stratified sample of 50 is taken. How many from each group?

Solution

Total 1000: 50 × 240/1000 = 12M1

12, 28, 10A1

Practice exercise 2

1. Identify the method: a researcher interviews the first 30 people leaving a cinema.

2. Give one reason why an online survey about internet use may be biased.

3 Reliability of data and outliers

Can we trust the data?

Data may be unreliable because of measurement error, recording mistakes, small samples or biased collection. An outlier is a value far from the rest. It may be a genuine extreme value or an error; investigate before removing it. The usual test is: x is an outlier if x < Q1 − 1.5 × IQR or x > Q3 + 1.5 × IQR.

WORKED EXAMPLE 2 Outlier test (standard)

Q1 = 24, Q3 = 38. Is 62 an outlier?

Solution

IQR = 14; upper fence 38 + 21 = 59M1 A1

62 > 59: outlierR1

Practice exercise 3

1. Q1 = 12, Q3 = 20. Find the fences.

2. A recorded height of 17.5 m in a class of 16-year-olds is an outlier. What should be done?

4 Frequency tables and grouped data

Organizing data.

A frequency table lists values or classes with their frequencies. For continuous data, classes such as 10 ≤ t < 20 have boundaries 10 and 20 and mid-interval value 15. The modal class has the highest frequency.

Time t (min)	0 ≤ t < 10	10 ≤ t < 20	20 ≤ t < 30	30 ≤ t < 40	40 ≤ t < 50
Frequency	6	14	22	13	5

Practice exercise 4

1. Write down the modal class and the mid-interval of the last class in the table above.

5 Histograms and frequency polygons

Pictures of distributions.

A histogram displays continuous grouped data with touching bars. In AI the classes are usually equal width, so the height is the frequency. The shape tells us about the distribution: symmetric, positively skewed (long right tail) or negatively skewed (long left tail).

Figure 1. Histogram of commuting times for 60 people.

Practice exercise 5

1. Describe the shape of the histogram.

2. Sketch a positively skewed distribution and state which is larger, the mean or the median.

6 Cumulative frequency graphs

Median, quartiles and percentiles.

Cumulative frequency is the running total. Plot it against the upper class boundary and join the points with a smooth curve or straight lines. Read the median at n/2, Q1 at n/4, Q3 at 3n/4 and the pth percentile at p% of n.

Figure 2. Cumulative frequency graph with Q1, median and Q3 at 15, 30 and 45.

WORKED EXAMPLE 3 Reading the graph (standard)

Estimate the median, IQR and the number taking longer than 35 minutes.

Solution

Median (30th) ≈ 24.5 minA1

Q1 (15th) ≈ 16.4, Q3 (45th) ≈ 32.3: IQR ≈ 15.9A1

cf at 35 ≈ 48.5: about 11–12 people take longerM1 A1

Practice exercise 6

1. Estimate the 90th percentile from the graph.

7 Box-and-whisker plots

The five-number summary.

A box plot shows minimum, Q1, median, Q3, maximum. Outliers are marked separately, and the whisker ends at the most extreme non-outlier. Box plots are ideal for comparing distributions side by side.

Figure 3. Box plots for two groups; 52 in group A is an outlier.

Practice exercise 7

1. Draw a box plot for 3, 5, 6, 8, 9, 10, 12, 14, 30, identifying any outlier.

8 Measures of central tendency

Mean, median, mode.

The mean uses all the data but is affected by outliers. The median is the middle value of the ordered data (position (n + 1)/2) and resists outliers; it suits skewed data such as incomes. The mode is the most common value; it is the only average for categorical data.

WORKED EXAMPLE 4 Effect of an outlier (standard)

Salaries ($ thousands): 32, 35, 36, 38, 40, 41, 250. Find the mean and median, and say which is more representative.

Solution

Mean = 472/7 = 67.4A1

Median 38A1

The median: the mean is pulled up by the outlier 250R1

Practice exercise 8

1. Find the mean, median and mode of 4, 6, 6, 7, 9, 10, 12.

2. Five numbers have mean 8. Four of them are 5, 7, 9, 12. Find the fifth.

9 Estimating the mean of grouped data

Use mid-interval values.

WORKED EXAMPLE 5 Grouped mean (standard)

Estimate the mean commuting time from the frequency table.

Solution

Σfx = 30 + 210 + 550 + 455 + 225 = 1470M1

Mean ≈ 1470/60 = 24.5 minA1

This is an estimate because the exact times within each class are unknown.

10 Measures of dispersion

How spread out?

The range (max − min) is simple but sensitive to outliers. The interquartile range (Q3 − Q1) measures the spread of the middle half and resists outliers. The standard deviation measures typical distance from the mean.

11 Variance and standard deviation

Technology does the arithmetic.

In AI, the standard deviation is found with a GDC (σx). Interpret it: if two classes have similar means but class A has σ = 4 and class B has σ = 11, class B's scores are much more spread out.

GDC tip

Enter values in L1 (and frequencies in L2), run 1-Var Stats L1, L2. Read x̄, σx, n, minX, Q1, Med, Q3, maxX. Use mid-interval values for grouped data.

WORKED EXAMPLE 6 Standard deviation (standard)

Find the mean and standard deviation of the commuting data.

Solution

GDC with mid-intervals: x̄ = 24.5A1

σ = 10.9 minA1

Practice exercise 9

1. Find the standard deviation of 12, 15, 15, 18, 20, 22.

2. Explain what a standard deviation of 0 means.

12 Effect of constant changes

Shifting and scaling data.

Adding a constant k to every value adds k to the mean, median and quartiles, but leaves range, IQR and standard deviation unchanged. Multiplying every value by k multiplies all of these measures by k (and the variance by k2).

WORKED EXAMPLE 7 Unit conversion (standard)

Times have mean 24.5 min and SD 10.9 min. Find the mean and SD in seconds, and after adding a 2-minute walk to every commute.

Solution

Seconds: mean 1470 s, SD 654 sA1

Add 2 min: mean 26.5 min, SD 10.9 min (unchanged)A1

Practice exercise 10

1. Each score is multiplied by 1.2 then 5 is added. The original mean was 60 and SD 8. Find the new mean and SD.

13 Comparing distributions

Centre, spread, shape, outliers.

A good comparison refers to: a measure of centre (median or mean), a measure of spread (IQR or SD), the shape, and any outliers, always in the context of the data.

WORKED EXAMPLE 8 Comparing two groups (exam level)

Group A: median 31, IQR 8. Group B: median 28, IQR 10. Compare the groups' test scores.

Solution

Group A scored higher on average (median 31 vs 28)R1

Group B's scores were more variable (IQR 10 vs 8)R1

Practice exercise 11

1. Two machines fill bottles. Machine P: mean 500.2 ml, SD 1.1 ml. Machine Q: mean 501.5 ml, SD 3.4 ml. Which is better?

14 Bivariate data and scatter diagrams

Two variables together.

Bivariate data pair two variables for each individual. A scatter diagram shows the relationship. Describe direction (positive/negative), strength (strong/weak) and form (linear/non-linear), and note any outliers.

Figure 4. Sleep against reaction time for 8 students, with the regression line.

15 Pearson's correlation coefficient

Strength of linear association.

Pearson's r (−1 ≤ r ≤ 1) measures the strength and direction of a linear relationship. Values near ±1 indicate strong linear correlation; near 0, little linear correlation. Correlation does not imply causation.

\|r\|	Description
0 – 0.25	very weak
0.25 – 0.5	weak
0.5 – 0.75	moderate
0.75 – 1	strong

WORKED EXAMPLE 9 Pearson's r (standard)

Find r for the sleep data and interpret it.

Solution

GDC: r = −0.998A1

Very strong negative linear correlation: more sleep is associated with faster reactionsR1

16 The regression line

Least squares.

The regression line of y on x, y = ax + b, is found with technology and passes through the mean point (x̄, ȳ). Interpret the gradient as the change in y per unit increase in x. Use it to predict y within the data range (interpolation); predictions outside it (extrapolation) are unreliable.

WORKED EXAMPLE 10 Regression (exam level)

Find the regression line for the sleep data, interpret the gradient and predict reaction time for 7.2 hours.

Solution

GDC: y = −29.2x + 568A2

Each extra hour of sleep is associated with about 29 ms faster reactionR1

x = 7.2: 358 ms (interpolation, reliable)A1

Practice exercise 12

1. Would you use the line to predict reaction time for 12 hours of sleep? Explain.

17 Spearman's rank correlation

Monotonic relationships.

Spearman's rs is Pearson's r applied to the ranks of the data. It measures how well the relationship is described by a monotonic (always increasing or always decreasing) function, and is less affected by outliers. Tied values get the average of their ranks.

Judge 1 rank	1	2	3	4	5	6
Judge 2 rank	2	1	4	3	6	5

WORKED EXAMPLE 11 Spearman (standard)

Two judges rank six dancers as shown. Find rs and interpret.

Solution

Pearson's r on the ranks (GDC): rs = 0.829A1

Strong agreement between the judgesR1

Practice exercise 13

1. When might Spearman's rs be preferred to Pearson's r?

2. Find rs for x = 1, 2, 3, 4 and y = 2, 4, 8, 16, and explain why rs = 1.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 12 Mixed problem (exam level)

Ten values have mean 15 and standard deviation 3. A value of 26 is added. Find the new mean and decide whether 26 is more than 3 SDs above the old mean.

Solution

New mean = (150 + 26)/11 = 16M1 A1

15 + 3 × 3 = 24 < 26: yesR1

Practice exercise 14

1. The heights of 20 plants have mean 34 cm. Five more plants with mean 40 cm are added. Find the new mean.

19 Combining data sets and weighted means (HL)

Pooling groups.

The mean of a combined data set is the weighted mean: (n1x̄1 + n2x̄2)/(n1 + n2). Weighted means also combine assessment components with different weights, or prices in an index.

WORKED EXAMPLE 13 Weighted grade (standard)

A course grade is 25% coursework (score 72), 75% examination (score 64). Find the overall score.

Solution

0.25 × 72 + 0.75 × 64M1

= 66A1

Practice exercise 15

1. Class A (20 students) has mean 68; class B (30 students) has mean 74. Find the overall mean.

20 Non-linear regression and the coefficient of determination (HL)

Choosing the best model.

For non-linear relationships, the GDC fits quadratic, cubic, exponential, power and sinusoidal models. The coefficient of determination R2 (0 to 1) gives the proportion of the variation in y explained by the model; for linear regression, R2 = r2. Compare R2 between models, but also consider the context and behaviour beyond the data.

WORKED EXAMPLE 14 Comparing models (exam level)

For a data set, linear R2 = 0.86, quadratic R2 = 0.97, exponential R2 = 0.98. Which model?

Solution

Exponential has the highest R2R1

Choose exponential if the context supports constant percentage growth; check residualsR1

Practice exercise 16

1. Explain why R2 = 0.64 for a linear model when r = −0.8.

21 Investigation: is there a link between sleep and reaction time?

A Paper 3 style exploration.

1. Design a fair method to collect data on sleep and reaction time from 30 students, describing the sampling method. 2. Using the sample data above, find r, rs and the regression line. 3. Interpret the gradient and intercept, and state whether the intercept is meaningful. 4. Identify a possible confounding variable. 5. Explain why a strong correlation does not prove that sleep causes faster reactions.

Answers. 1. Random sampling from the school roll; consistent online reaction test. 2. r = −0.998; rs = −1; y = −29.2x + 568. 3. Gradient: about 29 ms faster per hour; the intercept (0 hours) is extrapolation and not meaningful. 4. Caffeine, age, screen time. 5. Correlation shows association; an experiment would be needed for causation.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

Data: 14, 18, 21, 22, 25, 27, 29, 31, 35, 58.

(a) Find the median and quartiles. [3]

(b) Show that 58 is an outlier. [2]

Markscheme

Part	Answer	Marks
(a)	median 26	A1
	Q1 = 21, Q3 = 31	A2
(b)	IQR 10; fence 31 + 15 = 46	M1
	58 > 46	R1
(c)	28	A1
	11.6	A1

Examiner's comments. GDC quartile methods may differ slightly; accept consistent answers.

Question 2 (Paper 2 style) (about 15 minutes)

The table shows advertising spend x ($ thousands) and sales y ($ thousands) for 7 months: x = 2, 3, 4, 5, 6, 7, 8; y = 41, 48, 52, 60, 63, 71, 75.

(a) Find Pearson's r and interpret it. [2]

(b) Find the regression line of y on x. [2]

(d) Estimate sales when x = 5.5 and comment on reliability. [2]

Markscheme

Part	Answer	Marks
(a)	r = 0.996	A1
	very strong positive linear correlation	R1
(b)	y = 5.68x + 30.2	A2
(c)	each extra $1000 on advertising is associated with about $5680 more sales	R1
(d)	61.4 thousand	A1
	reliable: interpolation with strong r	R1

Examiner's comments. Answers in context are required.

Question 3 (Paper 2 style) (about 10 minutes)

A stratified sample of 40 is taken from a school with 300 juniors and 500 seniors.

(a) Find the number of juniors in the sample. [2]

(b) Explain one advantage of stratified sampling here. [1]

Markscheme

Part	Answer	Marks
(a)	40 × 300/800	M1
	15	A1
(b)	both age groups are represented in proportion	R1

Examiner's comments. Proportional allocation is expected.

23 Common misconceptions and links

Students often think…	Actually…
the mean is always best	the median suits skewed data
correlation proves causation	it shows association
adding a constant changes the SD	it does not
r = 0 means no relationship	no linear relationship

How 4.2 connects to the rest of the course

Topic	Connection
4.5 Probability	relative frequency
4.7 Probability distributions	mean and variance
4.11 Inferential statistics	testing claims
2.2 Functions	regression models

24 Check your understanding

Section A: multiple choice

1. Height is

A. continuous B. discrete C. qualitative D. categorical

2. The IQR is

A. max − min B. median − Q1 C. Q3 − Q1 D. Q3 + Q1

3. r = 0.92 indicates

A. weak correlation B. strong positive linear correlation C. causation D. no correlation

4. Spearman's rs uses

A. raw values only B. frequencies C. means D. ranks

5. Predicting outside the data range is

A. interpolation B. extrapolation C. regression D. sampling

Section B: structured

6. Ages 12, 15, 15, 16, 18, 40: find the median and IQR. [2]

7. Explain why the regression line passes through (x̄, ȳ). [1]

Answers

1. A 2. C 3. B 4. D 5. B

6. 15.5; Q1 = 15, Q3 = 18: IQR 3

7. the least squares line always contains the mean point

25 Summary and topic map

Idea	Remember
Sampling	random, systematic, stratified, quota, convenience
Graphs	histogram, cumulative frequency, box plot
Outliers	beyond 1.5 IQR from quartiles
Centre	mean, median, mode
Spread	range, IQR, SD
Correlation	Pearson r (linear), Spearman rs (monotonic)
Regression	y on x through (x̄, ȳ); interpolate

Before the exam, make sure you…

•compare centre and spread in context
•investigate outliers
•state the technology used
•avoid causal claims

Descriptive Statistics on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 5. Topic map for Descriptive Statistics.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. (a) discrete quantitative; (b) qualitative; (c) continuous quantitative

2. a parameter describes a population; a statistic describes a sample

Practice exercise 2

1. convenience

2. it excludes people without internet access

Practice exercise 3

1. 0 and 32

2. it is almost certainly an error (17.5 cm or 1.75 m): correct or remove it

Practice exercise 4

1. 20 ≤ t < 30; 45

Practice exercise 5

1. roughly symmetric, slightly positively skewed

2. the mean

Practice exercise 6

1. 54th value: about 39 min

Practice exercise 7

1. Q1 = 5.5, median 9, Q3 = 13; upper fence 24.25: 30 is an outlier

Practice exercise 8

1. 7.71, 7, 6

2. 7

Practice exercise 9

1. 3.37

2. all values are equal

Practice exercise 10

1. 77; 9.6

Practice exercise 11

1. P: closer to 500 ml on average and more consistent

Practice exercise 12

1. no: extrapolation beyond the data

Practice exercise 13

1. for non-linear monotonic data or ranked data, or with outliers

2. the ranks are identical: y increases whenever x increases

Practice exercise 14

1. (680 + 200)/25 = 35.2 cm

Practice exercise 15

1. 71.6

Practice exercise 16

1. R2 = r2 = 0.64

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