Unit 01 β Descriptive Statistics
Summarisation of data: tabular (frequency table), graphical (histogram / frequency curve); introduction to inferential statistics; measure of central tendency β the arithmetic mean for a sample and a population.
Every question shape in this unit is drilled on Unit 01 β How to Solve Anything: five topics, each with a master method, ten fully worked questions and ten to attempt. Read the concepts here first, then go there.
Nothing computational. In both papers this unit appears only inside Q3 (a), where a table of summary statistics for two service channels is handed to you and you are asked to interpret it β starting with what the gap between mean, median and mode says about the shape of the distribution. Final Q3 (a) QP p. 2, Re-Exam Q3 (a) QP p. 2.
So learn this unit for vocabulary and reading, not for drawing pie charts. Both versions solved β
The deck is titled "Unit 1, 2 β Introduction n Summarization" but it stops at scatter diagrams slide 49. It never covers the arithmetic mean, which the syllabus puts in Unit 1 Syllabus p. 1. The mean is taught instead through the worked grouped-data problem in your class notes Class notes pp. 3β4 and Levin Β§3.2 p. 77. Both are used below.
1. What Statistics Is, and the Two Halves of It Core syllabus concept
1Understand the Concept
Statistics is the science of learning from data β collection, analysis and interpretation slide 2. It splits into two halves, and every later unit belongs to one of them:
| Descriptive Statistics | Inferential Statistics | |
|---|---|---|
| What it does | Collects, summarises and simplifies data that would otherwise be unwieldy and voluminous. | Uses a sample to generalise about the important aspects of a population. |
| When you use it | When the population of interest is small enough to describe directly. | When the population is too large to measure, so you must sample it. |
| The number it produces | A parameter if computed on the population. | A statistic if computed on the sample. |
| Where it lives on this site | Units 1β2 | Units 3β6 (and everything after) |
2Population and sample
- A population is the set of all elements about which we wish to draw conclusions.
- A sample is a subset of the elements of a population, selected to represent it slide 12.
- A variable is a characteristic or condition that can change or take different values slide 11.
3Why a manager cares
The deck gives a list you can lift straight into any "applications of statistics in management" answer slide 6: location of plant, size of plant, production planning, quality control, finance decisions, marketing decisions, personnel decisions, purchase decisions, sales decisions, accounting decisions. Slide 5 gives the professional version β economists reading unemployment and inflation, financial planners reading market trends, accountants sampling sales revenue, marketers reading consumer preference, production supervisors controlling quality, politicians reading opinion polls, physicians reading drug-effectiveness data slide 5.
2. Types of Data β and Why the Type Decides the Chart Core syllabus concept
1Understand the Concept
Every summarisation decision in this unit follows from one question: is the data qualitative or quantitative? Get that wrong and you will reach for the wrong table and the wrong chart.
Continuous: Measured on a continuous scale (e.g., weight, height, speed).
Rank / Ordinal: Ordered categories (e.g., 1st, 2nd, 3rd rank).
Taken from slides 8β10.
2Where the data comes from
| Primary data | Secondary data | |
|---|---|---|
| Definition | Do not already exist in any form; collected for the first time from the primary source. | Already exist in some form β published or unpublished β in an identifiable secondary source. |
| Effort | Fresh, first-time collection covering the whole population or a sample drawn from it. | Generally available from a published source, though not necessarily in the form actually required. |
slide 7. The caveat "not necessarily in the form actually required" is the standard exam point about secondary data β it is cheap but may not fit your question.
3Classification
Classification is "the process of arranging data into sequences and groups according to their common characteristics, or separating them into different but related parts" β Secrist slide 14. The four methods to name:
- Qualitative β dichotomy (two-fold, e.g. male / female) or manifold (nested, e.g. male-employed / male-unemployed / female-employed / female-unemployed) slide 17.
- Geographical β by state, city, region or zone (agricultural output per hectare by country) slide 19.
- Chronological / temporal β by hour, day, week, month, year (population of India by decade) slide 20.
- Variable β by discrete values (height in cm β number of students) or continuous class intervals (income βΉ1000β1500 β number of employees) slide 22.
3. Frequency, Relative and Percent Frequency Distributions Core syllabus concept
1Understand the Concept
A frequency distribution is a tabular summary of data showing the frequency (number of items) in each of several non-overlapping classes. Its whole purpose is to "provide insights about the data that cannot be quickly obtained by looking only at the original data" slide 25.
Three tables come out of the same counts:
2Worked example β Marada Inn
Twenty guests rated their accommodation as Poor / Below Average / Average / Above Average / Excellent slide 26. The three tables are one table:
| Rating | Frequency | Relative freq. | Percent freq. |
|---|---|---|---|
| Poor | 2 | .10 | 10 |
| Below Average | 3 | .15 | 15 |
| Average | 5 | .25 | 25 |
| Above Average | 9 | .45 | 45 |
| Excellent | 1 | .05 | 5 |
| Total | 20 | 1.00 | 100 |
Frequencies slide 27, relative and percent slide 30.
3The reading β this is what the exam actually wants
The deck spells out the two insights a manager takes from the pie chart of that table slide 35, and this is exactly the register Q3 (a) is written in:
- One-half of the customers surveyed gave a rating of "above average" or "excellent" β this might please the manager.
- For each customer who gave "excellent", two gave "poor" β this should displease the manager.
4. Choosing the Right Chart Core syllabus concept
1The decision table β memorise this one page
The deck's closing slide is a complete decision tree slide 49. Written out:
| Data | Tabular methods | Graphical methods |
|---|---|---|
| Qualitative | Frequency distribution Β· Relative frequency distribution Β· Percent frequency distribution Β· Crosstabulation | Bar graph Β· Pie chart |
| Quantitative | Frequency distribution Β· Relative frequency distribution Β· Cumulative frequency distribution Β· Cumulative relative frequency distribution Β· Stem-and-leaf display | Histogram Β· Ogive Β· Scatter diagram |
2The four charts, and the one detail each is examined on
| Chart | Built for | The detail that earns the mark |
|---|---|---|
| Bar graph | Qualitative data | The bars are separated, to emphasise that each class is a separate category slide 31. |
| Histogram | Quantitative data | The bars touch, because the classes are adjacent intervals on a continuous scale. |
| Pie chart | Relative frequency of qualitative data | A class with relative frequency .25 takes .25 Γ 360 = 90Β° of the circle slide 33. |
| Ogive | Cumulative frequency | Plotted at the upper class boundary, and the median is read off where it crosses n/2. |
3Stem-and-leaf β the one that keeps the actual numbers
A stem-and-leaf display shows both the rank order and the shape of a distribution. It is "similar to a histogram on its side, but it has the advantage of showing the actual data values" slide 37. First digits go left of the vertical line (the stem); the last digit of each item goes right, in rank order (the leaves).
Hudson Auto Repair β 50 tune-up parts costs 5 | 2 7 6 | 2 2 2 2 5 6 7 8 8 8 9 9 9 7 | 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 8 | 0 0 2 3 5 8 9 9 | 1 3 7 7 7 8 9 10 | 1 4 5 5 9 Leaf unit = 1 (so "5 | 2" means 52)
slides 40β41. If the leaf unit is 0.1, "8 | 6" means 8.6 slide 39. Where the leaf unit is not shown, assume it equals 1 slide 38.
4Scatter diagram β reading the relationship
A scatter diagram shows the relationship between two quantitative variables, one per axis. Three patterns to name on sight slides 46β48: positive (rises left to right), negative (falls), no apparent relationship (a cloud).
- Drawing a histogram for qualitative data. Ratings, brands and regions get a bar graph, with gaps.
- Forgetting that classes must be non-overlapping. "2.0 to 2.5" and "2.5 to 3.0" is ambiguous for a value of exactly 2.5.
- Reading causation from a scatter diagram. The Panthers slide says associated with, not caused by.
- Choosing too many classes. Levin shows the same 20 inventory figures in 6 classes and in 12 classes p. 20 β the 12-class version is mostly empty cells and hides the pattern.
5. The Arithmetic Mean β Sample and Population Core syllabus concept
1Understand the Concept
The mean is the syllabus's named measure of central tendency for Unit 1 Syllabus p. 1. Two formulas, and the symbol tells the marker which one you are using:
The grouped formula is the one your class actually worked Class notes p. 3; Levin Β§3.2 p. 77.
2The rule your lecturer wrote in red
"For normal calculations, an inclusive series need NOT be converted to an exclusive series for calculating the MEAN. However for calculating MEDIAN and MODE conversion IS REQUIRED." Class notes p. 3, in red
An inclusive series is written 1β3, 4β6, 7β9 β there is a gap between 3 and 4. To make it exclusive you subtract 0.5 from each lower limit and add 0.5 to each upper limit, giving 0.5β3.5, 3.5β6.5, 6.5β9.5. The midpoints are unchanged (2, 5, 8), which is exactly why the mean does not care.
3Worked example β the Georgia hospital problem
This is the problem your class solved in full Class notes pp. 3β4. The administrator of a Georgia hospital surveyed the number of days 200 randomly chosen patients stayed in hospital following an operation.
- Step 1 β Lay out the classes and find the midpoints
The midpoint of an inclusive class is (lower + upper) / 2. For 1β3 that is (1+3)/2 = 2.
Hospital stay (days) Midpoint x Frequency f f Β· x 1β3 2 18 36 4β6 5 90 450 7β9 8 44 352 10β12 11 21 231 13β15 14 9 126 16β18 17 9 153 19β21 20 4 80 22β24 23 5 115 Total β Ξ£f = 200 Ξ£fx = 1543 - Step 2 β Apply the grouped mean formula ΞΌ = Ξ£fx / Ξ£f = 1543 / 200 = 7.7 days
- Step 3 β State it in words
The average post-operative stay is about 7.7 days. Your notes then take Ο = 4.6 for the same data and go on to apply Chebyshev's theorem β that continuation is worked in Unit 2 Concept 5.
4Median and mode β needed to read Q3 (a)
The exam never asks you to compute these, but it hands you all three and asks what their ordering means, so you need the definitions:
- Median β the middle value once the data is arranged in order; half the observations lie below it. Levin Β§3.5 p. 96.
- Mode β the most frequently occurring value. A distribution can have two (bimodal); the Final's Call Centre channel had a mode of "4 and 5" QP p. 2. Levin Β§3.6 p. 104.
Mean = Median = Mode β symmetric
Mean < Median < Mode β negatively skewed (a left tail of low scores drags the mean down)
Mean > Median > Mode β positively skewed (a right tail of high values drags the mean up)
This is developed properly in Unit 2 Concept 3.
Practice A1βA6 β grouped mean, class intervals and chart choice β