DP Maths AA HL - Statistics and Probability - Descriptive Statistics

DP Maths AA HL - Statistics and Probability - Descriptive Statistics - Full notes

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DP Maths AA HL | Descriptive Statistics

Descriptive Statistics

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4.1 Descriptive Statistics

DP MATHEMATICS AA · TOPIC 4: STATISTICS AND PROBABILITY

4.1

Sampling and Descriptive Statistics

Full notes for HL · Syllabus SL 4.1, 4.2, 4.3

Contents

1	Populations, samples and data
2	Sampling methods and bias
3	Presenting data: frequency tables and histograms
4	Cumulative frequency graphs
5	Box-and-whisker plots and outliers
6	Measures of central tendency
7	Measures of dispersion
8	Effect of constant changes on statistics
9	Designing a statistical investigation
10	Types of data and appropriate displays
11	Grouped data: estimating statistics
12	Percentiles and percentile ranks
13	Outliers: identifying and handling
14	Comparing distributions
15	Linear transformations of data
16	Combining data sets
17	Variance and standard deviation by hand
18	Mixed review: worked problems
19	Additional exam-style practice
20	Sampling: random numbers and stratification
21	Exam-style questions with markscheme
22	Common misconceptions and links
23	Check your understanding
24	Summary and topic map
25	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 Quick check.

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 from samples, and discrete from continuous data
describe sampling techniques (simple random, convenience, systematic, quota, stratified) and identify bias
construct and interpret frequency tables, histograms and cumulative frequency graphs
estimate the median, quartiles and percentiles from a cumulative frequency graph
draw box-and-whisker plots and identify outliers using 1.5 × IQR
calculate the mean, median and mode, including from grouped data
calculate the range, interquartile range, variance and standard deviation using technology
describe the effect of adding or multiplying all data values by a constant

This booklet covers SL 4.1, 4.2, 4.3, which is common to SL and HL.

Guiding questions

How can large data sets be summarized fairly? How can the method of collecting data affect the conclusions we draw?

How this topic is examined

Statistics questions appear on both papers but are mostly Paper 2, with a GDC. Typical tasks: read a cumulative frequency graph, compare two distributions using box plots, calculate statistics from grouped data, identify outliers, and comment on sampling methods.

When comparing distributions, always compare a measure of centre and a measure of spread, 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
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
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 on Paper 2
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); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 Populations, samples and data

The language of statistics.

Think about it

A school wants to know how long its 1200 students sleep each night. Is it necessary to ask every student? What could go wrong if only one class is asked?

A population is the entire group being studied; a sample is a subset of the population from which data are collected. Statistics calculated from a sample are used to estimate population parameters.

Data may be discrete (countable values, such as the number of siblings) or continuous (measured on a scale, such as height or time). Continuous data are often grouped into classes. Outliers are values far from the rest of the data; they should be investigated and only removed if they are errors.

The reliability of conclusions depends on the sample size and on how representative the sample is.

2 Sampling methods and bias

Representative samples.

Method	How it works	Strength / weakness
Simple random	every member has an equal chance (random numbers)	unbiased; needs a full list
Convenience	uses easily available members	quick; likely biased
Systematic	every kth member from a list after a random start	easy; biased if the list has a pattern
Quota	fixed numbers from each group, chosen non-randomly	matches proportions; not random
Stratified	random sample from each group in proportion to its size	representative of subgroups; needs group sizes

Bias occurs when the sampling method systematically favours some outcomes, for example surveying only people at a gym about exercise habits.

WORKED EXAMPLE 1 Stratified sample (standard)

A school has 400 students in Year 12 and 350 in Year 13. A stratified sample of 60 is taken. How many students from each year?

Solution

Total 750; Year 12: 60 × 400/750M1

= 32; Year 13: 60 × 350/750 = 28A1

Practice exercise 1

1. Identify the sampling method: choosing every 20th name on the register after a random start.

2. Explain why surveying shoppers at 10 a.m. on a weekday about working hours is biased.

3. A company has 120 managers, 480 office staff and 900 factory staff. Find the numbers in a stratified sample of 50.

3 Presenting data: frequency tables and histograms

Grouping continuous data.

A frequency table lists each value or class with its frequency. For grouped data, the mid-interval value represents each class in calculations. A histogram shows grouped continuous data with bars touching; in this course classes have equal widths, so the bar height is the frequency.

Figure 1. A histogram of the homework times of 50 students.

Time t / hours	0 ≤ t < 10	10 ≤ t < 20	20 ≤ t < 30	30 ≤ t < 40	40 ≤ t < 50
Frequency	5	12	18	10	5
Mid-interval	5	15	25	35	45

WORKED EXAMPLE 2 Mean from grouped data (standard)

Estimate the mean homework time for the data above.

Solution

Σfx = 25 + 180 + 450 + 350 + 225 = 1230M1

Mean ≈ 1230/50 = 24.6 hoursA1

This is an estimate because the actual values within each class are unknown.

4 Cumulative frequency graphs

Running totals.

The cumulative frequency of a value is the number of data values less than or equal to it. Plot cumulative frequency against the upper class boundary and join with a smooth curve (or straight lines). From the graph, estimate the median (at n/2), the quartiles Q1 (at n/4) and Q3 (at 3n/4), and any percentile.

Figure 2. Cumulative frequency graph, with Q1, median and Q3 read off at 12.5, 25 and 37.5.

WORKED EXAMPLE 3 Reading the graph (standard)

Use the graph to estimate the median, the interquartile range and the 90th percentile.

Solution

Median (25th value): about 24.4 hoursA1

Q1 (12.5): about 16.3; Q3 (37.5): about 32.5; IQR ≈ 16.3M1 A1

90th percentile (45th value): 40 hoursA1

Practice exercise 2

1. Using the graph, estimate how many students spend more than 35 hours.

2. Find the percentile corresponding to 20 hours.

5 Box-and-whisker plots and outliers

The five-number summary.

A box-and-whisker plot shows the five-number summary: minimum, Q1, median, Q3, maximum. The box covers the middle 50% of the data. A value is an outlier if it lies more than 1.5 × IQR below Q1 or above Q3; outliers are marked with a cross and the whisker stops at the most extreme non-outlier.

Box plots are ideal for comparing distributions: compare medians (centre), IQRs (spread) and shape.

Figure 3. Box plots comparing two classes; the score 48 in class A is an outlier.

WORKED EXAMPLE 4 Outlier test (standard)

Class A has Q1 = 18.5, Q3 = 27 and maximum 48. Determine whether 48 is an outlier.

Solution

IQR = 8.5; 1.5 × IQR = 12.75M1

Upper fence 27 + 12.75 = 39.75A1

48 > 39.75, so 48 is an outlierR1

Practice exercise 3

1. Data: 3, 5, 7, 8, 9, 11, 12, 13, 30. Find the quartiles and determine whether 30 is an outlier.

2. Compare the two distributions in the box plot above.

6 Measures of central tendency

Mean, median and mode.

The mean uses every value but is affected by outliers. The median (middle value when ordered; position (n + 1)/2) is resistant to outliers and better for skewed data. The mode is the most frequent value; for grouped data the modal class is the class with the highest frequency.

WORKED EXAMPLE 5 Finding a missing value (exam level)

The mean of 5, 8, x, 12, 2x is 9. Find x.

Solution

(25 + 3x)/5 = 9M1

3x = 20, x = 20/3A1

Practice exercise 4

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

2. Ten numbers have mean 6. One number is removed and the mean becomes 5.5. Find the number removed.

7 Measures of dispersion

Range, IQR, variance, standard deviation.

The range is max − min (affected by outliers). The interquartile range IQR = Q3 − Q1 (the spread of the middle half). The variance is the mean of the squared deviations from the mean, and the standard deviation σ is its square root, in the same units as the data:

In this course, the standard deviation is found using technology; you should be able to interpret it. A larger standard deviation means the data are more spread out about the mean.

GDC tip

Use the one-variable statistics function, entering values in one list and frequencies in another. Read off x̄, σx, n, min, Q1, Med, Q3, max. Use σx (the population standard deviation) unless told otherwise.

WORKED EXAMPLE 6 Standard deviation from a table (standard)

Find the mean and standard deviation of the number of goals: 0 goals (3 games), 1 (5), 2 (7), 3 (4), 4 (1).

Solution

n = 20, Σfx = 0 + 5 + 14 + 12 + 4 = 35, mean = 1.75A1

Σfx2 = 0 + 5 + 28 + 36 + 16 = 85; variance = 85/20 − 1.752 = 1.1875M1

σ = 1.09A1

Practice exercise 5

1. Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

2. Estimate the standard deviation of the homework data using mid-interval values.

8 Effect of constant changes on statistics

Shifts and scales.

If a constant b is added to every value, measures of centre (mean, median, mode, quartiles) increase by b, but measures of spread (range, IQR, σ) are unchanged. If every value is multiplied by a constant a > 0, measures of centre and spread are multiplied by a; the variance is multiplied by a2.

Change to each value	Mean, median	SD, IQR, range	Variance
x + b	+ b	unchanged	unchanged
ax	× a	× a	× a2
ax + b	× a, then + b	× a	× a2

WORKED EXAMPLE 7 Converting units (exam level)

Temperatures have mean 20 °C and standard deviation 4 °C. Find the mean and standard deviation in °F, using F = 1.8C + 32.

Solution

Mean = 1.8(20) + 32 = 68 °FA1

SD = 1.8 × 4 = 7.2 °FA1

Practice exercise 6

1. Each of a set of scores is increased by 5. The original mean was 62 and SD 8. State the new mean and SD.

2. Each value is doubled. The variance was 9. Find the new variance.

9 Designing a statistical investigation

From question to conclusion.

A good investigation states a clear question, defines the population, chooses a sampling method that avoids bias, collects data reliably, summarizes them appropriately, and draws conclusions that acknowledge limitations.

WORKED EXAMPLE 8 Planning (standard)

A school wants to know whether Year 12 students sleep less than Year 10 students. Suggest a sampling method and two statistics to compare.

Solution

Stratified random sample from each year groupR1

Compare medians (or means) and IQRs (or standard deviations)R1

Practice exercise 7

1. Identify a source of bias in asking students to record their sleep in a class discussion.

2. Why is a larger sample usually better?

10 Types of data and appropriate displays

Match the display to the data.

Data type	Example	Suitable displays
categorical	favourite sport	bar chart, pie chart
discrete	number of siblings	bar chart, frequency table
continuous	height	histogram, cumulative frequency, box plot
bivariate	height and mass	scatter diagram

WORKED EXAMPLE 9 Choosing a display (standard)

Which display would best compare reaction times for two groups?

Solution

Parallel box plots (or two histograms on the same scale)R1

11 Grouped data: estimating statistics

Mid-intervals and interpolation.

For grouped data, use mid-interval values to estimate the mean and standard deviation, and linear interpolation within a class to estimate the median and quartiles: median ≈ L + ((n/2 − F)/f) × w, where L is the lower boundary of the median class, F the cumulative frequency before it, f its frequency and w its width.

Mass (kg)	40–50	50–60	60–70	70–80	80–90
Frequency	4	11	18	12	5

WORKED EXAMPLE 10 Median by interpolation (exam level)

Estimate the median mass.

Solution

n = 50; 25th value in 60–70; F = 15, f = 18M1

60 + (10/18)(10) = 65.6 kgA1

WORKED EXAMPLE 11 Mean and SD (standard)

Estimate the mean and standard deviation.

Solution

Mid-intervals 45, ..., 85: mean = 65.6A1

SD (GDC) = 10.8 kgA1

Practice exercise 8

1. Estimate the lower quartile of the mass data by interpolation.

12 Percentiles and percentile ranks

Position within a distribution.

The pth percentile is the value below which p% of the data lie. From a cumulative frequency graph, read across at p% of n. The percentile rank of a value is the percentage of data at or below it.

WORKED EXAMPLE 12 Percentile (standard)

For 200 test scores, the cumulative frequency at 72 marks is 158. Find the percentile rank of 72.

Solution

158/200 = 0.79M1

79th percentileA1

Practice exercise 9

1. Explain the meaning of 'the 90th percentile of income is $85 000'.

13 Outliers: identifying and handling

1.5 × IQR and context.

A value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3. Outliers should be investigated: remove them only if they are errors. Report statistics with and without outliers when appropriate.

WORKED EXAMPLE 13 Effect of an outlier (standard)

Data: 12, 14, 15, 15, 16, 17, 18, 45. Find the mean and median with and without 45.

Solution

With: mean 19, median 15.5A1

Without: mean 15.3, median 15A1

The mean is strongly affected; the median hardly changesR1

Practice exercise 10

1. Q1 = 30, Q3 = 42. Find the range of values that are not outliers.

14 Comparing distributions

Centre, spread, shape, outliers.

A full comparison mentions a measure of centre (median or mean), a measure of spread (IQR or SD), the shape (symmetric or skewed), and any outliers, always in context.

WORKED EXAMPLE 14 Comparison (exam level)

Class A: median 64, IQR 12, symmetric. Class B: median 70, IQR 25, positively skewed with an outlier at 100. Compare.

Solution

B has a higher median: on average B scored higherR1

B's scores are more spread out and skewed, with one very high scoreR1

Practice exercise 11

1. Two machines fill bags: A has mean 501 g, SD 1.2 g; B has mean 503 g, SD 4.5 g. Which is better if the target is 500 g?

15 Linear transformations of data

Effect on statistics.

If each value x is transformed to ax + b: mean, median and quartiles become a(old) + b; standard deviation, IQR and range are multiplied by |a| (unaffected by b); the variance is multiplied by a2.

WORKED EXAMPLE 15 Scaling marks (standard)

Test scores have mean 52 and SD 8. Each is transformed by y = 1.25x + 5. Find the new mean and SD.

Solution

Mean 1.25(52) + 5 = 70A1

SD 1.25 × 8 = 10A1

Practice exercise 12

1. Temperatures in °C have median 18 and IQR 6. Find them in °F (F = 1.8C + 32).

16 Combining data sets

Pooled mean.

The mean of combined groups is the weighted mean (n1x̄1 + n2x̄2)/(n1 + n2). Combined standard deviations require Σx2 for each group: Σx2 = n(σ2 + x̄2).

WORKED EXAMPLE 16 Combined mean (standard)

Group A (12 students) has mean 68; group B (18 students) has mean 74. Find the combined mean.

Solution

(816 + 1332)/30M1

= 71.6A1

Practice exercise 13

1. 10 values have mean 20 and SD 3; 15 values have mean 25 and SD 4. Find the combined mean and SD.

17 Variance and standard deviation by hand

Understanding the formula.

σ2 = Σ(x − x̄)2/n = Σx2/n − x̄2. The second form is quicker by hand. The standard deviation is the square root of the variance and has the same units as the data.

WORKED EXAMPLE 17 By hand (standard)

Find the SD of 3, 5, 6, 8, 13.

Solution

x̄ = 7; Σx2 = 303A1

σ2 = 60.6 − 49 = 11.6M1

σ = 3.41A1

Practice exercise 14

1. Find the SD of 2, 4, 4, 4, 5, 5, 7, 9.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 18 Mixed problem (exam level)

Five numbers have mean 10, median 9, mode 8 and range 7. The numbers are integers. Find them.

Solution

Sorted a, b, 9, d, e with mode 8: a = b = 8 (8 appears twice)M1

e − 8 = 7: e = 15; sum 50: d = 10A1

8, 8, 9, 10, 15A1

Practice exercise 15

1. The mean of x, 2x, 3x, 10 and 15 is 11. Find x and the median.

19 Additional exam-style practice

Paper 2.

WORKED EXAMPLE 19 Paper 2 question (exam level)

The times (min) for 60 runners: 20–25: 4; 25–30: 13; 30–35: 21; 35–40: 15; 40–45: 7. (a) Estimate the mean and SD. (b) Estimate the median. (c) The fastest 20% receive a medal. Estimate the qualifying time.

Solution

(a) mean 33.2, SD 5.44A1 A1

(b) 30th value: 30 + (13/21)(5) = 33.1M1 A1

Practice exercise 16

1. Find the IQR of the running times by interpolation.

20 Sampling: random numbers and stratification

Doing it in practice.

Simple random sampling: number the population and use a random number generator. Systematic sampling: every kth member after a random start. Stratified: sample from each stratum in proportion to its size. Quota and convenience sampling are quicker but more prone to bias.

WORKED EXAMPLE 20 Stratified (standard)

A company has 150 managers, 600 office staff and 1250 factory workers. Find the numbers in a stratified sample of 80.

Solution

Total 2000A1

6, 24, 40A1

Practice exercise 17

1. Describe how to choose a simple random sample of 30 from 450 students.

21 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 2 style) (about 15 minutes)

The times, in minutes, taken by 80 runners to complete a race are summarized: 20 ≤ t < 25: 6; 25 ≤ t < 30: 18; 30 ≤ t < 35: 30; 35 ≤ t < 40: 16; 40 ≤ t < 45: 10.

(a) Write down the modal class. [1]

(b) Estimate the mean and standard deviation. [3]

(d) The fastest 10% of runners receive a medal. Estimate the maximum time for a medal. [2]

Markscheme

Part	Answer	Marks
(a)	30 ≤ t < 35	A1
(b)	mid-intervals 22.5, ..., 42.5	M1
	mean 32.9	A1
	sd 5.52	A1
(c)	cf: 6, 24, 54, 70, 80	A1
	40th value lies in 30 to 35	M1
	median ≈ 32.7	A1
(d)	8th value: 25 + (2/18)(5)	M1
	≈ 25.6 min	A1

Examiner's comments. Linear interpolation within a class gives estimates; graph readings within tolerance are accepted.

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

Two groups of plants were grown with different fertilizers. Heights (cm): Group P: Q1 = 14, median 18, Q3 = 21, min 9, max 26. Group Q: Q1 = 16, median 22, Q3 = 25, min 11, max 41.

(a) Determine whether any value in group Q is an outlier. [3]

(b) Compare the two groups. [3]

Markscheme

Part	Answer	Marks
(a)	IQR = 9; 1.5 × 9 = 13.5	M1
	upper fence 38.5	A1
	41 > 38.5: outlier	R1
(b)	Q has a higher median (22 vs 18): taller on average	R1
	Q has a larger IQR (9 vs 7): more variable	R1
	conclusion in context: fertilizer Q gives taller but less consistent growth	R1

Examiner's comments. Comparisons must refer to both centre and spread, in context.

22 Common misconceptions and links

Students often think…	Actually…
the median is the middle value without ordering	order the data first
outliers must always be removed	investigate; remove only if errors
adding a constant changes the SD	it does not
a grouped mean is exact	it is an estimate

How 4.1 connects to the rest of the course

Topic	Connection
4.4 Correlation	bivariate data
4.5 Probability	relative frequency
4.9 Normal distribution	mean and standard deviation
4.14 Continuous RVs (HL)	median and mode of a distribution

23 Check your understanding

Section A: multiple choice

1. A sample chosen as every 10th item is

A. systematic B. simple random C. stratified D. quota

2. The IQR is

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

3. An outlier lies beyond

A. 2 × SD B. 1.5 × IQR from the quartiles C. the range D. the median

4. Adding 10 to each value changes

A. the SD B. the IQR C. the range D. the mean

5. Height is

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

Section B: structured

6. Explain one advantage of stratified sampling over simple random sampling. [1]

7. The mean of 8 values is 12 and the SD is 3. Each value is multiplied by 2 and then 5 is subtracted. Find the new mean and SD. [2]

Answers

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

6. Ensures each subgroup is represented in proportion.

7. Mean 19; SD 6.

24 Summary and topic map

Idea	Remember
Sampling	random, convenience, systematic, quota, stratified; bias
Presentation	histograms, cumulative frequency, box plots
Centre	mean Σfx/Σf, median, mode
Spread	range, IQR, variance, σ
Outliers	beyond Q1 − 1.5 IQR or Q3 + 1.5 IQR
Transformations	x + b shifts centre; ax scales centre and spread

Before the exam, make sure you…

•compare centre and spread in context
•use mid-intervals for grouped data
•use the GDC for σ
•justify outliers with the fences

Sampling and 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 4. Topic map for Sampling and Descriptive Statistics.

25 Answers to practice exercises

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

Practice exercise 1

1. systematic

2. people with full-time jobs are mostly at work, so they are under-represented

3. 4, 16, 30

Practice exercise 2

1. 50 − 40 = 10 students

2. 17/50 = 34th percentile

Practice exercise 3

1. Q1 = 6, Q3 = 12.5 (GDC method); upper fence 22.25: 30 is an outlier

2. A has a higher median and similar spread; A has an outlier

Practice exercise 4

1. 8.17, 8, 7

2. 60 − 49.5 = 10.5

Practice exercise 5

1. 2

2. 11.1 hours

Practice exercise 6

1. 67; 8

2. 36

Practice exercise 7

1. social pressure to give typical answers; recall bias

2. estimates are more reliable (less sampling variability)

Practice exercise 8

1. 12.5th value: 50 + (8.5/11)(10) = 57.7 kg

Practice exercise 9

1. 90% of incomes are at or below $85 000

Practice exercise 10

1. 12 to 60

Practice exercise 11

1. A: closer to target and more consistent

Practice exercise 12

1. 64.4 °F; 10.8 °F

Practice exercise 13

1. mean 23; Σx2 = 4090 + 9615 = 13 705: SD = √(548.2 − 529) = 4.38

Practice exercise 14

1. 2

Practice exercise 15

1. 6x + 25 = 55: x = 5; data 5, 10, 10, 15, 15: median 10

Practice exercise 16

1. Q1 (15th): 29.2; Q3 (45th): 37.3: IQR 8.1

Practice exercise 17

1. number students 1–450; generate 30 distinct random integers in this range

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