Descriptive Statistics: Data Collection

• Course notes for APPM1022A are intended to complement lectures and other course materials for Introductory Statistics for Construction.
• Students should consult references for additional material and views.
• This material is under development and feedback is appreciated.

Descriptive Statistics

• Descriptive statistics, probability, and inferential statistics describe statistics.
• Descriptive statistics uses techniques and measures for collecting, organizing, and making sense of data.
• Descriptive statistics also involves turning data into meaningful information like graphs, charts, tables, and summary numerical measures.
• Probability aids decision-makers in measuring uncertainty.
• Inferential statistics involves drawing conclusions from a sample of large data.

Data Collection

• The main statistical role is providing decision-making methods by obtaining and converting data into useful information.
• Qualitative data provides descriptions, such as emotions or perceptions, and uses explorative methods to gain insights and motivations.
• Quantitative data provides quantifiable data for mathematical calculation or statistical analysis.
• Quantitative data answers questions such as how many, how often, or how much.

Data Levels

• Variables can be defined using four levels: nominal, ordinal, interval, and ratio.
• Nominal: Weakest data measurement technique; names variables without order (e.g., eye color).
• Ordinal: Ranks variables in order without determining the difference between them (e.g., happy, unhappy)
• Interval: Numerical variables with known, equal intervals (e.g., time).
• Ratio: Variables with measurable intervals (e.g., weight).

Methods of Collecting Data

• A population is a collection of all items of interest.
• A sample is a subset of a population, chosen randomly or methodically
• A sample must be representative of the population.
• Population is denoted by N, sample is denoted by n.
• Census: Measurement of the entire population, used by governments.
• Using a sample instead of a census is more economical, time-saving, and scientifically sound.
• Observations: Gathering data by observing people, requires trained observers.
• Time and differing perceptions are major considerations in using observation as a data collection form

Personal and Telephonic Interviews

• Interviews are cheaper than observations
• Interviews can use closed-ended questions, which are statistically easier to analyze or open-ended questions, which provide a much better response.
• Interviewer phrasing can influence answers, and sensitive topics may not get honest answers.
• Telephonic interviews rapidly gather data at a lower cost than personal observations.
• The disadvantages include the possibility of a reluctance to answer or easy call termination.

Questionnaires and Contemporary Methods

• Questionnaires are the cheapest data collection method; they eliminate interviewer bias, but may have poor response rate.
• Contemporary methods for data collection include: links, mentimeter, webcams, bluetooth, drones, social media monitoring, online tracking, GPS, wireless/web-based technologies, satellites, spacecrafts bar codes, handwaves, facial/voice recognition technologies and sensing technologies.
• It is very important to consider challenges, errors, and costs when choosing collection methods.

Methods of Representing Data

• Organizing collected data is important in order to give it meaning.
• Data from a hardware store shows item sale prices can be organized through a frequency distribution table.
• Categorize data into classes to create a frequency distribution table.
• Determine the number of classes using formula 2^k >= n where n is the total number of data values and solving for k will give the number of classes.
• Determine class width and the class width formula is class width = (largest value – smallest value) / number of values
• Develop a frequency distribution by counting observations that belong to each class.
• Determine the midpoint of each class with midpoint = (lower class limit + upper class limit) / 2.
• Compute the cumulative frequency by summing the frequency of a class and all preceding classes.

Presenting Data

• Pie charts are effective in representing data, particularly for budgetary allocations, due to their ease of understanding.
• The total is subdivided, and the pieces are proportional to the amounts they represent.
• Bar graphs, or rod diagrams, use vertical and horizontal bars to represent data along axes.
• Each bar represents a single value, making bar charts straightforward and effective.
• Histograms: Visual representation of frequency distribution where the number of observations per class are represented by the height of the bar
• Time series plots represent data measured over time and are useful for identifying changes in variables.
• Scatter plots identify relationships between two variables.

Charts

• Ensure data isn't misrepresented through scaling differences or starting axes from non-zero values.

Computer Applications

• Use statistical computer packages like Microsoft Excel, SPSS, or ANOVA
• Microsoft Excel is a spreadsheet program used to organize, manipulate, and analyze all kinds of data.
• Microsoft Excel is also versatile in presenting data visually and to draw diagrams.
• It is important to both represent and understand data using computer applications
• Media, politicians, and business misrepresent data to try and persuade the public, and get customers to buy their products.

Descriptive Statistics Introduction

• Statistics involves methods for collecting, organizing, summarizing, presenting, and analyzing data to make reasonable decisions.
• Collecting data on characteristics of a group may be impossible
• Instead of examining the population, one can examine a sample.
• If a sample represents a population, sample analysis could infer important conclusions about the larger population.
• Inductive statistics or statistical inference deals with such conditions used to draw the conclusions, probability is also used in stating such conclusions.
• Descriptive or deductive statistics seeks only to describe and analyse a given group without drawing any conclusions or inferences about a larger group

Important Statistical Concepts

• A variable is a symbol like X that assumes values from its domain; a constant has only one value.
• A continuous variable can assume any value between two points, whereas a discrete variable cannot

Discrete Variables

• The number of children in a family that can assume the value 0,1,2..., but not and rational number is a discrete variable.
• The age A of an individual which can be 50 years or 50.8 years depending on measurement accuracy, is a continuous variable.
• The number of children in family is example of discrete data, with heights of students as an example of continuous data.
• Measurements usually yield heights, whereas enumeration yields discrete data

Frequency Distributions

• Raw data: Unorganized numerical data
• Array: An arrangement of raw data in ascending or descending order
• Range: The difference between the smallest and largest data values
• Frequency distributions: Method of distributing the data into classes/categories to determine each classes frequency.
• A tabular arrangement of data classes with their class frequencies is a frequency distribution or table.
• The first category in a frequency table is called first class.

Ungrouped Data

• Data is represented in a list.

Measures of Central Tendencies

• Sample Mean: x = ( Σ Xi ) / n
• Sample Mode: Most frequently appearing value in a dataset, can be multi-modal if there are multiple modes
• Sample median: Arrange data values in ascending or descending order.
• Sample median: If odd numbered, the median is the middle number, but if it is even, then the median is the average of the two middle values

Measures of Variability

• Range: The range of a data set, R, is defined by formula
• Range formula: R = Xmax - Xmin
• A smaller range = less variability, larger range = opposite

Variance

• Variance measures how much a set of data points are dispersed around their mean value.
• Population variance (sigma squared) = sum of squared data differences between observed values and population mean, divided by the total number of observations.
• By squaring, always get non-negative computations, which indicates data dispersion, and distance. Dispersion cannot be negative.
• Squaring also amplifies the effect of large differences.
• Sample variance (s squared) = the sum of squared differences between observed sample values and the sample mean, divided by the number of samples, minus one
• Sample variance formula: s2 = Σ(Χ; −x)2 / n-1

Standard Deviation

• The square root of variance and its result is known as standard deviation.
• Standard deviation is more meaningful than variance
• Sample standard deviation formula: s= √ Σ(Χ – x)2 / n-1

Coefficient of Variation

• The coefficient of variation is the standard deviation divided by the mean.
• Another name: relative standard deviation.
• The population and sample formula can be calculated based on the availability.
• Comparing standard deviations of 2 data sets is meaningless, comparing coefficients of variation of 2 data is not

Coefficient of Skewness

• Data is subdivided into two by the median, and can be described as symmetric or asymmetric.
• In symmetric distribution, right and left sides are minor images, the curve would be represented by bell-shaped curve
• The data set not symmetrical is asymmeteric, and may be skewed.
• Measuring skewness can be done by evaluating how the measures of central tendencies relate to one another.
• If the mean, median, and mode match, the distribution is symmetrical. If they dont match, the skewness is in either the from left to the right.
• The distribution has a long trail extending to the right, it is skewed to the right, and positively skewed.

Types of Data Sets

• For a positively skewed data set:
  • Mean > Mode (Always)
  • Median > Mode (Always)
  • Mean > Median (Most of the time)
• For a left-skewed distribution, the is less than than the median
• Pearson's coefficient measures the strength and direction of skewness with statistics, standard deviation, mean and mode.
• There coefficients of skewness are as follows:
• α3= n / (n-1)(n-2)s3 Σ(Χ-x)3
• α3 = x - mode / standard deviation
• α3 = 3(x-median) / standard deviation

Skewness Implications

• If Mean < Median < Mode, distribution negatively skews left, coefficients will be negative.
• If the mean, median and mode values are equal, has to be a normal distribution and the coefficient is 0.
• When having few data points, the median is favored as a measure of central tendencies.
• Exercise 2.2.1 is an example of how to determine skewness of a data set

Measures of Position

• In statistics, percentiles, quartiles and deciles analyse and give meaning to data
• It shows where a value is compared to the rest of the data set.
• Pk= k(n) / 100, with n indicating total number of a data set arranged in ascending order. It shouldnt be mistaken for precentage.
• Quartiles split data (ascending order) to give different insights.

Permutations and Combinations - Basic Principle of Combinatorial Analysis

• A combination is a selection of items or events, in which we disregard order
• Combinatorial analysis includes events.
• nPr = n(n-1)… (n − r + 1) =n! / (n-r)! is the general formula (4.10) to counting subsets of elements
• Note that . nPn = n!.

Grouped Data Measures

• Grouped data is arranged in a frequency table as such distinct values of x are shown in the first column, and frequency representing its value, f, indicates the amount of value x shown in the column. (section 2.3)
• Mean of grouped data: ( Σ=1 Xi fi) / (Σ fin)
• Standard deviation:2 = Σ(Χ; −x)2 / n-1