Introduction to Statistics

Statistics is the science of collecting, organising, analysing and interpreting data to make decisions under uncertainty. A computer scientist uses it for performance measurement, A/B tests, machine learning, quality control and research.

1.1 Meaning, scope and limitations

In the singular sense statistics is a subject (a method of studying data); in the plural sense it means numerical facts (the statistics of exam results).

Two branches: descriptive statistics summarises the data you have (mean, charts); inferential statistics draws conclusions about a population from a sample (estimation and testing).

Population is the complete set of units of interest; a sample is a part of it; a number describing a population is a parameter (\mu ,\ \sigma), a number computed from a sample is a statistic (\bar{x} ,\ s).

Scope: business and economics (demand forecasting), industry (quality control), government (census, planning), health (clinical trials), computing (benchmarking, network traffic modelling, data science), and research in every science.

Limitations: statistics deals with aggregates, not individuals; it studies quantitative facts (qualitative facts must be converted to numbers); results are true on average, not for every case; it can be misused (selective data, misleading graphs, wrong sampling); and its results are only as good as the data. "There are lies, damned lies, and statistics" is a warning about misuse, not about the method.

Levels of measurement: nominal (labels), ordinal (order but no meaningful differences), interval (equal differences, no true zero: temperature in ^{\circ} C), ratio (true zero, ratios meaningful: income, weight).

The level decides which statistics make sense: you can compute a mean of ratio data but only a mode for nominal data.

By time: cross-sectional data (many units at one time, e.g. today's marks of all students), time-series data (one unit over time, e.g. monthly rainfall), panel data (both).

1.4 Methods of collecting primary data

• Direct personal investigation (interview): reliable and detailed; costly; interviewer bias.
• Indirect oral investigation: questioning people who know about the subject.
• Questionnaire / schedule: a questionnaire is filled by respondents (mailed or online), a schedule by an enumerator. Good questions are short, clear, neutral, in logical order; pilot-test them.
• Observation and experiment; telephone and web surveys; local reports and correspondents.

Problems of collection: non-response; bias from leading questions or unrepresentative samples; inaccurate answers (memory, sensitivity); cost and time; ambiguous definitions; enumerator errors.

Secondary data should be checked for source reliability, the definitions used, the date, and the accuracy and units. Editing (checking completeness and consistency) comes before tabulation.

1.5 Seeing the structure of data

(a) Hollerith census machine, replica at the Computer History Museum

(b) The five steps of a statistical study

1.6 Eight reasons why every computing student needs statistics

1. Measuring performance honestly. One run of a program says little, because timings vary from run to run. The mean and standard deviation of many runs tell you the typical speed and how stable it is. A Nepali web team comparing two hosting plans would use exactly this before choosing one.
2. Making decisions under uncertainty. Inferential statistics lets you decide from a sample while admitting the chance of error. A bank such as Nabil or NIC Asia tests a new mobile-banking screen on 2,000 users before showing it to millions. The result carries a measured confidence instead of a guess.
3. Machine learning and data science. Regression, classification and clustering are statistical ideas in code. Without knowing variance, bias and sampling, a student cannot tell a good model from a lucky one. Nepal's Election Commission and NRB data sets are natural practice material.
4. Quality control in industry. Factories sample a few items from each batch and judge the whole batch from them. A Biratnagar noodle mill or a Bhaktapur brick kiln uses this to keep defects low at low cost. Software teams do the same with bug counts per release.
5. Government planning and the census. The National Statistics Office counts people, houses and farms to plan schools, hospitals and roads. The 2078 BS census used handheld digital collection, so statistics and computing met in one project. Policies such as the allocation of seats and budgets depend on these figures.
6. A/B testing and product research. Two versions of an a p p screen are shown to two random groups, and a statistical test says whether the difference is real. Fonepay and eSewa-type wallets run such experiments on button text and offers. Only a test, not opinion, shows what actually works.
7. Health, agriculture and environment. Clinical trials, crop-yield trials and river-pollution monitoring all depend on sampling and comparison. A trial of a new paddy variety in the Terai compares yields on many plots, not one. The same logic protects patients and farmers from false claims.
8. Reading the news critically. Graphs with cut axes, tiny samples and selective percentages are common in advertisements and politics. A student who knows what the sample was, how it was chosen and what the margin of error is can spot a weak claim at once. This is the everyday use of the subject.

(a) Florence Nightingale (statistical graphics)

(b) Anscombe's quartet: same numbers, four pictures

(a) A ballot drop box

(b) Sample size for 95 percent confidence, p = 0.5

Formula and fact sheet

• Descriptive v s inferential; population/parameter v s sample/statistic.
• Qualitative: nominal, ordinal. Quantitative: discrete, continuous. Scales: nominal, ordinal, interval, ratio.
• Primary: first-hand. Secondary: existing sources.

Practice

1. Differentiate descriptive and inferential statistics; population and sample; parameter and statistic.
2. Classify: blood group, number of bugs, server response time, exam grade, temperature in ^{\circ} C.
3. State four limitations of statistics.
4. Compare primary and secondary data.
5. What is the difference between a questionnaire and a schedule?
6. List four problems in data collection.

Model answers: medium (5 marks) and long (10 marks) questions

Q1 (5 marks). Distinguish between primary and secondary data. Describe two methods of collecting primary data.

Introduction. Every statistical study starts with data, and the first question is where the data came from. Primary data are collected first-hand by the investigator for the present purpose.

Secondary data were collected earlier by someone else for another purpose and are reused.

Comparison (Table).

Two collection methods. (1) Direct personal interview: the enumerator asks each respondent a schedule of questions. It gives detailed and reliable answers and allows clarification, but it is costly and the interviewer may bias answers.

(2) Questionnaire (online or mailed): respondents fill in the form themselves. It is cheap and reaches many people, as with a Google Form sent to alumni, but non-response is high and nobody can explain a confusing question.

Conclusion. Use primary data when no suitable record exists; use secondary data first when it does, and always check its definitions and date.

Q2 (5 marks). Explain the four scales of measurement with one example each. Why does the scale matter?

Introduction. The scale of measurement describes how much information a variable carries, and it decides which statistics are legal to compute.

The four scales.

1. Nominal: labels with no order, such as blood group or district. Only counting and the mode make sense.
2. Ordinal: labels with a meaningful order but unequal gaps, such as grade A, B, C or a satisfaction rating. The median and percentiles are allowed; averaging the codes is doubtful.
3. Interval: equal differences but no true zero, such as temperature in ^{\circ} C. The mean and standard deviation are valid, but 40 ^{\circ} C is not "twice as hot" as 20 ^{\circ} C.
4. Ratio: equal differences and a true zero, such as income, weight or response time. Every statistic and ratio is valid: 60 ms is twice as slow as 30 ms.

Why it matters. The scale picks the average (mode, median, mean), the chart (bar chart for nominal, histogram for ratio) and the test. Averaging student roll numbers is arithmetic but statistical nonsense, because roll numbers are only labels.

Conclusion. Identify the scale first; the right method follows from it.

Q3 (10 marks). Define statistics. Explain its scope and limitations, and describe the sources and methods of collecting data.

Introduction. In the singular sense, statistics is the science of collecting, organising, presenting, analysing and interpreting data to support decisions under uncertainty.

In the plural sense it means the numerical facts themselves, such as exam results.

Its two branches are descriptive statistics, which summarises the data in hand, and inferential statistics, which draws conclusions about a population from a sample (Figure 1.1 and Plate 1.1 show the data types and the five-step process).

Scope (six areas).

1. Business and economics: demand forecasting, price indices, risk.
2. Industry: sampling inspection and quality control.
3. Government: census, planning, budgets, election rolls.
4. Health and agriculture: clinical and crop trials.
5. Computing: benchmarking, network-traffic models, A/B tests, machine learning.
6. Research: every science tests ideas with data.

Limitations (five points).

1. Statistics deals with aggregates, not with the individual case.
2. It handles only quantitative facts; qualitative facts must first be coded.
3. Results are true on average, not in every instance.
4. It can be misused by selective data, truncated graphs or biased samples.
5. Conclusions are only as good as the data collected.

Sources of data. Primary data are collected first-hand (survey, experiment); secondary data come from existing records (census, NRB bulletins, company files). Primary data fit the need but cost more; secondary data are cheap but may be outdated.

Methods of collecting primary data. Direct personal interview, indirect oral investigation, questionnaire or schedule, observation, experiment, and telephone or web surveys. Good questions are short, clear, neutral and pilot-tested.

Typical problems are non-response, leading questions, memory errors and enumerator bias; collected forms are edited for completeness and consistency before tabulation.

Comparison (Table).

Conclusion. Statistics turns raw numbers into evidence, but only when the data are collected carefully and the limits of the method are respected.