Frequency Distribution & Qualitative Data

Frequency Distribution

• It is a table showing how often different categories in qualitative datasets occur in order to organize and summarize data for analysis
• Frequency distribution construction involves collecting data through surveys, listing all possible qualitative data categories, counting category occurrences, and creating a two-column table for categories and frequencies

Qualitative Data Example

• Qualitative Data Exampe: Student's preferred mode of transport, listed as bus, bicycle, car, or motorcycle

Qualitative Data Example

• National Quality of Life Survey collects data from Colombian households
• (ECV) Level of Education Categories: no education, primary, secondary, technical/technological, and university
• Political Culture Survey assesses Columbians' perceptions and attitudes toward politics and civic engagement
• Categorized by level of trust in institutions as very trustworthy, trustworthy, somewhat trustworthy, or not at all trustworthy
• Television programs categorized as types of programs like News, soap operas, sports, series, documentaries
• National Household Survey (ENH) investigates demographic and socio-economic character
• Categorized by type of dwelling such as a house, apartment, or rural residence

Graphical Representations: Bar Graphs

• Bar graphs compare frequencies across categories where each category is represented by height equivalent to its frequency

Graphical Representations: Pie Charts

• Pie charts display the proportion of each category in relation to the whole dataset, making each category a "slice" of the total pie

Pareto Charts

• Pareto charts combine bar and line graphs.
• Bars display the frequency of categories in descending order, while the line shows the cumulative percentage

Frequency Distribution (Quantitative)

• It is a table displaying observed numerical values
• Values are distributed across different class intervals
• The number of observations helps organize, summarize, and interpret

Frequency Distribution Construction

• Data collection is achieved via surveys
• Determine range by subtracting the minimum data from the maximum data
• Decide on the number of classes (intervals)
• Calculate class width by dividing the range by the number of classes

Frequency Distribution Example

• Construction permit areas in square meters
• Example Data Values: 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400
• Calculated range: 350
• A calculated number of classes results in 4
• Calculated class width: 88

Graphical Representation Example: Absolute Frequency Histogram

• Histograms using absolute frequency represent the absolute frequency of classes, using bars where height correlates with class frequency

Graphical Representation Example: Relative Frequency Histogram

• Relative frequency histograms correlate the frequency ratio with the bar heights

Graphical Representation Example: Frequency Polygon

• Frequency Polygon: It plots the midpoint of classes as a point and connects each point with a line that approximates the distribution
• It shows the entire data set

Graphical Representation: Dot Plot

• Dot plots visualize data in terms of density

Graphical Representation: Line Chart

• Line Chart: Plots data that represents values collected and analyzed to track trends

Stem and Leaf Plots

• Plots organize numerical data to show distribition
• They include stems and leaves, where stems are shown in the initial digits of the numerical data and leaves represent the final numbers

Stem and Leaf Plot Construction

• Ascending order
• Separating each number into the stem (digits excluding the last one) and leaf (last digit)
• The list must be shown in descending order
• Corresponding leaves are added in the equivalent stem on the row

Stem and Leaf Plot Example I

• Exam scores example: 78, 85, 90, 92, 65, 88, 73, 84, 91, 93, 70, 66, 89, 95, 87
• Stem and leaf plot presents exam scores ranging from 65 to 95
• The bulk of the scores concentrate in the 80-90 range, which reveals strong overall student performance.
• Few scores are in the lower and mid-range which shows students went beyond minimum standards
• Distribution skews to the right, this shows that a signficant number have high scores

Stem and Leaf Plot Example II

• Temperatures: 25,27,30,29,26,28,31
• Stem and leaf that shows Temperatures between 25 and 31 degrees
• Showing Temperatures in the 30s is considered not normal

Measures of Central Tendency

• These describe where a bulk of the points are located in a data distribution
• Mean: Average of numbers
• Median: The mid point between the min and max
• Mode: Most frequent occuring

Central Tendency Measurements for Ungrouped Data

• The central tendency measurements describe data by showing core data
• The principle tools are arithmetric mean, median, and mode

Arithmetic Mean

• Arithmetic Mean is the average in a data set

Properties of Arithmetic mean

• Uniqueness: Each data set has only one mean
• Simplicity: easy to find and calculate
• All data points are used to calculate the mean
• It is effected by extremes

Arithmetic Mean Example

• Student exam scores: 75, 80, 85, 90, 95
• Arithmetic mean calculation of notes resulted in mean score of 85

Weighted Mean

• This applies when numbers are more senstive for weight

Weighted Mean Formula

• Xp= (ΣXiWi) / (ΣWi) Xp denots mean, Wi represents dataset number total value

Weighted Mean:

• Example: grades and weigh, and uses weighted mean formula

The Median

• It's the value that divides a dataset in ascending order

Odd Data

• Odd data has median midpoint

even data

• Even data mid point is in between the 2 middle pont

Median example

• Set of class notes: 75, 80, 85 ,90,95
• The medial data points are in ascending order
• The data set n=5, therefore the median is 85

Mode

• A set with one mode is called an a unimodal data set
• Bimodal sets have 2 modes
• Multimodal sets have multiple modes
• If numbers do not repeat mode is not viable

Contexts for Mean, Median and Mode in Colombian Data

• Family Income:
  • Arithmetic mean: Calculate the average Colombia Family income
  • Weighted Mean: Weigh income based on household members
  • Median: Determining midpoint
  • Mode: Finding the most common
• Saber 11 exam results:
  • Arithmetic Mean: Calculating average student score
  • Weighted Mean: Assigning test to subject importance
  • Median: Dividing the student group into 2 groups
  • Mode: determining the most frequent score
• Bogotá travel times:
  • Arithmetic Mean: Calculating average travel time
  • Weighted mean: Weigh times considering frequency of daily times

Measuring Variability in Datasets

• Variability or disperson tells how far data stretches or is separated
• RAnge, Variance, Standard deviation, and coefficent of variation are indicators

Range

• The range is the maximum point subtracted by the minitum point
• It describes how far it covers of an area

Example

• Student grades points: 75, 80, 85, 90,95
• Calculated 95-75= 20 points

Variance

• Measures spread within dataset
• High variance mean wide data spread, while low variance means data is tightly knit
• Population formula : x2 = Σ(Xi - M)/N
• Sample Formula
• s2 = Σ(Xi -X)2/n-1

Standard Deviation

• This measure data variance from mean data set and is represented in same units as raw data.

Formula

• Standard deviaiation = √variance
• To find how far data shifts from the numbers average deviation set

Example points:

• 75,80,85,90,95 points on tests were used to find how how scattered set wa
• Resulted in approximation of 7,91 or 7.91%. This finds that set averagely drifts

Coefficient of variance

• Describes ratio of standard deviation to average
• Cv =σ/μ * 100 or S/X* 100

formula

• Describes how deviated points in dataset
• Compares varied measurements