STA121: Descriptive Statistics I Lecture Notes

Course Overview and Identification

STA121: Descriptive Statistics I is a $2$ -unit course presented by Dr. Emmanuel Torsen, a Senior Lecturer and Head of the Department of Statistics at Modibbo Adama University, Yola. These lecture notes, documented on February $13$ , $2025$ , originate from the Faculty of Physical Sciences. The course serves as a comprehensive introduction to the fundamental scientific methods used for handling numerical information and making data-driven decisions.

Definition and Scope of Statistics

Statistics is defined as a scientific method encompassing the collection, organization, summarization, presentation, and analysis of data. It includes the process of drawing valid conclusions based on such analysis. More broadly, it is the science of collecting, organizing, presenting, analyzing, and interpreting data to support more effective decision-making processes. It deals specifically with information gathering, condensation of data into compact forms, and the measurement of variation or relationships between identical or similar phenomena. In research, statistics involves the estimation of population characteristics from samples, the design of experiments and surveys, and the testing of hypotheses regarding populations. It is fundamentally concerned with the analysis of information collected through sampling where variability is likely to occur.

Levels of Measurement in Statistical Data

Data measurement is categorized into four distinct levels, each providing a different degree of mathematical utility:

Nominal: This level represents categories that do not have a specific or inherent order. An example is the type of school, such as public or private.
Ordinal: This represents categories with a meaningful rank or order, but the intervals between them are not consistent or measurable. Class ranks serve as a primary example.
Interval: This level consists of numeric data where the intervals between values are meaningful, yet there is no true or absolute zero point. SAT scores are a typical example.
Ratio: This is the highest level of numeric data, possessing a true zero point which allows for meaningful comparisons of magnitude. Attendance rates are a common example of ratio data.

Areas of Statistical Application

Statistics applies to any field involving extensive numerical data. Key areas include:

Industry: Statistics assists business leaders in making decisions amidst uncertainty. It is applied in market research, feasibility studies, investment policies, quality control, personnel selection, economic forecasting, and auditing. Looking at historical data allows businesses to prepare for future uncertainties.
Biological Science: It is used to analyze crop yields under varying environmental conditions and fertilizers. It also studies animal responses to different diets to optimize resource application. Medicine and public health advancements rely heavily on statistical principles.
Physical Sciences: Statistical metrology aids the fields of astronomy, chemistry, geology, meteorology, and oil exploration. Decisions on mineral investments are made after statistical examination of resource samples.
Government: The government is the most significant producer and user of statistical data, requiring up-to-date information on revenue, expenditures, population, health, and defense to make effective policy decisions.
Other Fields: Statistics is also integral to education, health, finance, and the judiciary.

Principal Uses of Statistical Data

Statistical data serves several critical functions: it summarizes large bulks of numerical data into representative qualities like the mean, standard deviation, variance, and coefficient of variation. It allows for reasonable deductions and general conclusions under specific conditions. Furthermore, statistics is essential for planning, as it allows future predictions based on historical patterns. It reveals the nature and pattern of variations through numerical measurement and ensures that data representation is both clear and accessible.

Classification and Types of Statistical Data

Statistical data is broadly divided into two main categories: Qualitative (Categorical) and Quantitative (Numerical).

Qualitative Data describes attributes or characteristics, such as student gender or level of education. Quantitative Data involves numerical measures, such as graduation rates or test scores. Quantitative data is further subdivided into:

Discrete Data: These are items that are countable, such as the number of students present in a classroom.
Continuous Data: This data can take any value within a specific range, representing measurements such as weight or the time spent studying for an exam.

Primary Sources of Statistical Data

Primary data is information collected directly from the field for a specific research objective. It is used when no existing reliable information is available. The reliability of primary data depends on the collection method, timeliness, and the caliber of the personnel involved.

Advantages of primary data include the investigator's high level of confidence in the data, a personal appreciation for the collection challenges at every stage, the comprehensive nature of the resulting report, and the inclusion of specific definitions and schedules used for collection.

Disadvantages include that the process is highly time-consuming, expensive, requires significant manpower, and the data may become obsolete by the time it is published.

Secondary Sources of Statistical Data

Secondary data is information that has already been collected and published by other institutions, governments, or researchers. It is common in historical comparisons or large-scale analyses. Typical sources include ministry statistical divisions, banks, insurance companies, research institutions, and the print media. For example, vehicle registration data can be retrieved from the Board of Internal Revenue.

Advantages of secondary data are its low cost, efficiency in time (as compared to primary collection), and ease of availability.

Disadvantages include the risk of misuse or misinterpretation, potential difficulty in access due to official protocols, and the possibility that the data does not conform to the investigator\'s specific needs. Additionally, it is often hard to determine the precision of the data because the original collection method is unknown, and the data may contain transcription errors from the primary source.

Methods of Statistical Data Collection

There are four primary methods used to collect data:

Surveys and Questionnaires: This involves using specially designed forms to extract information from respondents (individuals, groups, or organizations). This method is widely used because it provides wide coverage, saves time, is cost-effective, and can collect diverse information types. However, challenges include the difficulty of designing clear, non-ambiguous questions, the risk of non-response, and the possibility of respondents providing inaccurate or deliberately false information.
Interview Method: This represents personal, oral interaction between an interviewer and respondent, either face-to-face or via telephone. It allows for detailed information gathering and full explanations, and it permits the interviewer to correct false information on the spot. Limitations include limited geographic coverage, the time-intensive nature of the process, high costs, the need for advance appointments, and the potential for personal feelings to bias the data.
Observation (Experiment) Method: This is a systematic, "on the spot" watching of an event as it happens. It is common in biological, social, and physical laboratory experiments. It provides high accuracy because events are recorded in real-time as they happen, ensuring a direct link between the observer and the subject matter. The observer can guide or control the process to maintain accuracy.
Documentary Method: This uses existing reports (books, newspapers, bulletins, official records) to predict future phenomena. Official documents provide official data, while others yield unofficial data. Advantages include extreme time and cost savings and the ability to access historical records. The main disadvantage is that if the original record was wrong, inaccurate, or incomplete, the current collection will carry those errors with no way to correct them.

Specific Sources of Documentary Data in Education

Documentary data can be retrieved from various specialized entities:

Government Publications: Includes the UNESCO Institute for Statistics (UIS) for global/regional education data, the National Bureau of Statistics (NBS) for national attainment and enrollment reports, and Ministry of Education reports for student-teacher ratios and infrastructure details.
International Organizations: The OECD (Organization for Economic Cooperation and Development) provides PISA (Programme for International Student Assessment) data. The World Bank Education Statistics and UNICEF Education Reports provide global participation and equity data.
Institutional Records: Schools and universities maintain records on attendance, graduation, dropout rates, and admission statistics.
Educational Databases: The National Center for Education Statistics (NCES) in the US (which conducts the NAEP survey), EdStats (World Bank), and the Educational Research Data Center (ERDC).
Academic Sources: Published articles in journals such as the Journal of Educational Psychology and various dissertations or theses.
Digital Platforms: Data from Massive Open Online Courses (MOOCs) like Coursera and edX, and analytics from Learning Management Systems (LMS) like Blackboard or Canvas.
Surveys and Assessments: The Annual School Census, national literacy surveys, and international assessments like TIMSS (Trends in International Mathematics and Science Study) and PIRLS (Progress in International Reading Literacy Study).
Web-Based Data: Social media platforms like Twitter, Facebook, and Reddit, and academic sites like Google Scholar and ResearchGate.

Principles of Data Presentation: Descriptive Statistics

Data comes from variables, which are characteristics that assume different values. Numerical (Quantitative) variables take numeric values (e.g., height), while Qualitative variables are designated by names. Raw data consists of unorganized collected measurements; once arranged in order of magnitude, it is called an array. A tabular arrangement showing the frequency of measurements or class intervals is known as a frequency distribution or frequency table.

Frequency Distribution Terminology

Class: A grouping of observations.
Class Limits: Includes the Lower class limit (lower bound) and Upper class limit (upper bound).
Class Interval: The range of values in a class (e.g., $20-29$ ). Closed intervals have defined limits; open intervals (e.g., " $20$ and above") do not.
Class Boundaries: Also called "true class limits." The upper boundary of a class is the average of its upper limit and the lower limit of the subsequent class.
Class Size (Width): The difference between the upper and lower class boundaries of a class.
Class Marks (Midpoints): The average of the upper and lower class limits.
Frequency: The actual number of observations in a specific class.
Relative Frequency: The ratio of a class frequency to the sum of all frequencies, often expressed as a percentage.
Cumulative Frequency: The total of all class frequencies up to and including a specific class.
Cumulative Frequency Distribution: A table showing the number of observations less than a given upper class boundary.

Construction of Grouped Frequency Distributions

Constructing a frequency distribution follows specific steps:

Identify the range (difference between the largest and smallest values).
Divide the range into a convenient number of non-overlapping class intervals. The class size is roughly the range divided by the number of classes.
Determine the count for each interval using tallies (grouped in bundles of $5$ ).

In Example $2.1$ , scores for $40$ mathematics students were organized into $8$ classes. With a range divided by $8$ , the class size was approximately $9$ . The first class was settled at $22-30$ with a frequency of $5$ . The final interval was $85-93$ with a frequency of $5$ .

Graphical Description of Statistical Data

Pie Chart: Best for qualitative data. It is a circle where sector angles are proportional to frequency. Angles are calculated by multiplying the relative frequency by $360^{\circ}$ .
Bar Chart: Uses separate vertical (or horizontal) bars of equal width. The height of the bar is proportional to frequency. Bars must be separated by equal gaps, generally about $1.5$ times the width of a bar. It is used for discrete, categorical, nominal, and ordinal data.
Component and Multiple Bar Charts: Used when data classes contain sub-classes (e.g., male vs. female enrollment).
Histogram: A graph where bars are erected on class boundaries with no gaps between them. The width corresponds to class size, and the height reflects the frequency. The center of each bar is the class midpoint.
Frequency Polygon: A line graph of frequencies against class marks. It is formed by connecting the midpoints of the tops of histogram bars with straight lines.
Ogive: Also known as a cumulative frequency curve. It is produced by plotting cumulative frequencies (less than) against upper class boundaries. This graph is essential for determining the median, quartiles, and percentiles.

Numerical Description: Measures of Central Tendency

These statistics summarize the center or location of measurements.

The Mean (Arithmetic Mean): The sum of measurements divided by the number of measurements ( $n$ ). For ungrouped data: $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ . For grouped data: $\bar{X} = \frac{\sum f_i X_i}{\sum f_i}$ . The mean is unique and takes every value into account, but it is sensitive to extreme values (outliers).
The Median: The middle score when data is arranged in order. For ungrouped data (if $n$ is odd): $Median = X_{\left(\frac{n+1}{2}\right)}$ . If $n$ is even: $Median = \frac{X_{\left(\frac{n}{2}\right)} + X_{\left(\frac{n}{2}+1\right)}}{2}$ . For grouped data: $Md = L_l + \left[ \frac{\frac{N}{2} - cf_b}{f_{Md}} \right] C$ , where $L_l$ is the lower boundary of the median class, $N$ is total frequency, $cf_b$ is cumulative frequency before the median class, $f_{Md}$ is the frequency of the median class, and $C$ is the class size.
The Mode: The measurement with the highest frequency. Data can be bimodal (two modes) or multimodal. For grouped data: $Mo = L_l + \left[ \frac{\Delta_1}{\Delta_1 + \Delta_2} \right] C$ , where $\Delta_1$ is the frequency difference with the preceding class and $\Delta_2$ is the difference with the following class. The mode can be found graphically by intersecting diagonals on the highest bar of a histogram and tracing to the horizontal axis.

Numerical Description: Measures of Dispersion

Dispersion measures the spread of data around a fixed point, usually the mean.

Range: The difference between the maximum and minimum values. It is simple but influenced heavily by extreme values.
Mean Deviation ( $MD$ ): The mean of the absolute deviations from the mean. $MD = \frac{\sum |X_i - \bar{X}|}{n}$ for ungrouped data, and $MD = \frac{\sum f_i |X_i - \bar{X}|}{\sum f_i}$ for grouped data.
Variance ( $S^2$ ): The average of the squared deviations from the mean. For ungrouped data: $S^2 = \frac{\sum (X_i - \bar{X})^2}{n-1}$ . For grouped data: $S^2 = \frac{\sum f_i (X_i - \bar{X})^2}{(\sum f_i) - 1}$ .
Standard Deviation ( $S$ ): The positive square root of the variance: $S = \sqrt{S^2}$ .
Coefficient of Variation ( $CV$ ): A measure of relative dispersion expressed as a percentage: $CV = \left( \frac{S}{\bar{X}} \right) \times 100$ . It is useful for comparing data sets with different units or significantly different means.

Practical Examples of Numerical Analysis

Example $3.8$ presents the daily electrical consumption in Jimeta, Adamawa State ( $2.9, 6.1, 6.9, 0.8, 3.5, 2.9$ in $10^7$ watts). The statistics calculated based on $n=7$ (as per the transcript's provided solution steps) are:

Mean ( $\bar{X}$ ): $4.3 \times 10^7 \text{ watts}$
Mean Deviation ( $MD$ ): $2.0 \times 10^7 \text{ watts}$
Variance ( $S^2$ ): $5.6 \times 10^{14} \text{ watts}^2$
Standard Deviation ( $S$ ): $2.4 \times 10^7 \text{ watts}$
Coefficient of Variation ( $CV$ ): $55.7\%$

Example $3.9$ calculates statistics for a frequency distribution of $135$ observations:

Mean ( $\bar{X}$ ): $90.24$
Mean Deviation ( $MD$ ): $13.90$
Variance ( $S^2$ ): $266.31$
Standard Deviation ( $S$ ): $16.32$
Coefficient of Variation ( $CV$ ): $18.08\%$