Chapter 2: Descriptive Statistics and Data Visualization Patterns and Data Organization and Visualization

Introduction to Descriptive Statistics

• Objective of Descriptive Statistical Methods: The primary aim is to present information and data in a manner that is clear, concise, and accurate.

• The Problem of Data Volume: Analyzing large data sets is difficult because there is often too much information for the human mind to assimilate or understand.

• Task of Descriptive Methods: These methods summarize information and extract main features without distorting the overall picture.

Organizing and Graphing Data

• Samples and Populations: All data sets used in this context are regarded as samples drawn from a specific population.

• Purpose of Samples: A sample is studied mainly to obtain information about the larger population.

• Focus of Study: The main goal involves summarizing and describing specific features of the data set.

• Raw Data (Ungrouped Data):

  • Definition: Information obtained from each member of a population or sample and recorded in the sequence in which it becomes available.

  • Characteristics: Collected at random; not organized or ranked.

Frequency Distributions

• Definition: A frequency distribution is a table where data are grouped into classes, and the number of values (frequencies) falling in each class is recorded.

• Purpose: To gain insight into the distribution pattern of frequencies across different classes; the name refers specifically to this pattern.

Example 1: Survey of Urban Neighborhood Families

• Context: A survey of $40$ families recorded the number of children per family.

• Sample Size ($n$): $40$

• Raw Data: $1, 0, 3, 2, 1, 5, 6, 2, 2, 1, 0, 3, 4, 2, 1, 6, 3, 2, 1, 5, 3, 3, 2, 4, 2, 2, 3, 0, 2, 1, 4, 5, 3, 3, 4, 4, 1, 2, 4, 5$

• Frequency Distribution Table Results:

  • $0$ children: Frequency ($f$) = $3$; Relative Frequency ($Rf$) = $3/40 = 0.075$

  • $1$ child: Frequency ($f$) = $7$; Relative Frequency ($Rf$) = $7/40 = 0.175$

  • $2$ children: Frequency ($f$) = $10$; Relative Frequency ($Rf$) = $10/40 = 0.25$

  • $3$ children: Frequency ($f$) = $8$; Relative Frequency ($Rf$) = $8/40 = 0.2$

  • $4$ children: Frequency ($f$) = $6$; Relative Frequency ($Rf$) = $6/40 = 0.15$

  • $5$ children: Frequency ($f$) = $4$; Relative Frequency ($Rf$) = $4/40 = 0.1$

  • $6$ children: Frequency ($f$) = $2$; Relative Frequency ($Rf$) = $2/40 = 0.05$

  • Total Frequency: $\sigma f = n = 40$

Relative Frequency Calculations

• Definition: The proportion (or percentage) of data falling into a specific class.

• Formula:   $Rf = \frac{\text{Class Frequency}}{\text{Sample Size}} = \frac{f}{n}$

• Note: The sum of frequencies always equals the sample size ($n$).

Example 2: Smartphone Ownership (Qualitative Data)

• Context: Type of smartphone owned by the youngest family member.

• Categories: $A$ = Android, $I$ = iPhone, $W$ = Windows phone.

• Raw Data: $A, W, I, I, W, I, A, A, A, I, W, A, W, A, I, A, I, I, A, W, W, I, A, A, I, W, A, I, A, I, W, I, I, W, A, I, A, A, A, W$

• Frequency Distribution Results:

  • Android: Frequency = $16$; $Rf = 0.4$; Cumulative Frequency ($F$) = $16$

  • iPhone: Frequency = $14$; $Rf = 0.35$; Cumulative Frequency ($F$) = $30$

  • Windows phone: Frequency = $10$; $Rf = 0.25$; Cumulative Frequency ($F$) = $40$

Constructing Frequency Distributions for Grouped Data

Case Study: Staff Lunch Spending at DUT

• Data Set: Amounts spent in Rands by $40$ lecturers.

• Values: $38, 64, 50, 32, 44, 25, 49, 57, 46, 58, 40, 47, 36, 48, 52, 44, 68, 26, 38, 78, 63, 21, 54, 65, 46, 73, 42, 47, 35, 53, 40, 35, 61, 45, 35, 42, 50, 56, 45, 28$

• Statistics: Highest value = $78$, Lowest value = $21$, $n = 40$.

Major Steps for Construction:

1. Calculate Number of Classes ($k$):

   • Formula: $k = 3.3\log(n) + 1$

   • Calculation: $k = 3.3\log(40) + 1 = 6.2868 \approx 6$

2. Calculate Class Width ($W$):

   • Formula: $W = \frac{\text{Range}}{k}$

   • Range ($R$): $78 - 21 = 57$

   • Calculation: $W = \frac{57}{6} = 9.5 \approx 10$

3. Choose Starting Point:

   • Rule: Use any convenient number equal to or less than the smallest value in the data set as the lower limit of the first class ($20$ was chosen for this example).

Grouped Frequency Distribution Table:

• Classes:

  • 20 - < 30: $f = 4$, $Rf = 4/40$, $F = 4$, Midpoint ($x$) = $25$

  • 30 - < 40: $f = 7$, $Rf = 7/40$, $F = 11$, Midpoint ($x$) = $35$

  • 40 - < 50: $f = 14$, $Rf = 14/40$, $F = 25$, Midpoint ($x$) = $45$

  • 50 - < 60: $f = 8$, $Rf = 8/40$, $F = 33$, Midpoint ($x$) = $55$

  • 60 - < 70: $f = 5$, $Rf = 5/40$, $F = 38$, Midpoint ($x$) = $65$

  • 70 - < 80: $f = 2$, $Rf = 2/40$, $F = 40$, Midpoint ($x$) = $75$

• Percentage Frequency: Calculated as $Rf \times 100$.

• Cumulative Frequency ($F$): The frequency of a class plus all previous frequencies. For the first class, $F$ always equals the frequency ($f$).

• Midpoint (Class Mark): The average of the two class limits.

  • Formula: $\text{Midpoint} = \frac{\text{lower class limit} + \text{upper class limit}}{2}$

  • Example: $M_1 = \frac{20 + 30}{2} = 25$

Exercises for Practice

Exercise 1: Call Center Performance

• Metric: Service level (time in seconds to answer calls).

• Data ($n = 50$): $16, 14, 16, 19, 6, 14, 15, 5, 16, 18, 17, 22, 6, 18, 10, 15, 12, 6, 19, 16, 16, 15, 13, 25, 9, 17, 12, 10, 5, 15, 23, 11, 12, 14, 24, 9, 10, 13, 14, 26, 19, 20, 13, 24, 28, 15, 21, 8, 16, 12$

• Task: Construct a frequency and cumulative distribution.

Exercise 2: Public Transport Spending

• Data: Amount spent per day by $50$ DUT staff members.

• Statistics: Highest = $R64$, Lowest = $R39$.

• Data Set: $57, 39, 52, 52, 43, 50, 53, 42, 58, 55, 58, 50, 53, 50, 49, 45, 49, 51, 44, 54, 49, 57, 55, 64, 45, 50, 45, 51, 54, 58, 53, 49, 52, 51, 41, 52, 40, 44, 49, 45, 43, 47, 47, 43, 51, 55, 55, 46, 54, 41$

• Task: Construct a frequency and cumulative distribution.

Graphing Grouped Data

Histogram

• Definition: Graphical representation of a frequency distribution.

• Construction:

  • Horizontal axis: Class boundaries (true class limits ensuring continuity).

  • Vertical axis: Frequencies, relative frequencies, or percentages.

  • Representation: Rectangular bars with class boundaries as the base and frequency as the height.

  • Gaps: There are no gaps between bars because they are drawn over class boundaries.

• Insight: From the DUT lunch spending histogram, it is visible that most staff members spent between $40$ and $50$ Rand.

Shape of a Distribution

• Purpose: Histograms describe the clustering pattern of data values.

• Normal Distribution: Bell-shaped; $\text{Mean} = \text{Median} = \text{Mode}$.

• Right Skewed: Tail extends to the right; \text{Mode} < \text{Median} < \text{Mean}.

• Left Skewed: Tail extends to the left; \text{Mean} < \text{Median} < \text{Mode}.

• Central Location Rules:

  • If data is normally distributed, use the Mean.

  • If data is not normally distributed, use the Median.

Frequency Polygon

• Definition: A line graph emphasizing continuous change in frequencies.

• Construction:

  • Plot class midpoints against frequencies.

  • Add two additional classes (one at each end) with zero frequency to anchor the graph to the horizontal axis.

  • Join adjacent points with straight lines.

• Equivalence: The histogram and frequency polygon are equivalent; the area under both represents the total number of observations ($n$).

Cumulative Frequency Graph (Ogive)

• Definition: A graph of cumulative frequencies versus upper-class boundaries.

• Construction:

  • Horizontal axis: Upper class boundary.

  • Vertical axis: Cumulative frequency ($F$).

  • Starting Point: Plot $0$ against the lower-class boundary of the first class.

  • Example coordinates: $(20, 0), (30, 4), (40, 11), (50, 25), (60, 33), (70, 38), (80, 40)$.

Graphing Qualitative Data Sets

Bar Charts

• Definition: Categorical data represented by rectangular bars.

• Features: Bars can be vertical or horizontal. Height/length is proportional to the size of the category. Only totals are represented.

• Example (Facebook Users, US 2011):

  • $13 - 25$ Age Group: $65,082,280$ users ($45\%$

  • $26 - 44$ Age Group: $53,300,200$ users ($36\%$

  • $45 - 64$ Age Group: $27,885,100$ users ($19\%$

  • Total users: $146,267,580$

• Example (Commerce Student Distinctions):

  • $0$ As: $2$ students; $1$ A: $6$ students; $2$ As: $9$ students; $3$ As: $4$ students; $4$ As: $3$ students.

Pie Chart

• Definition: A circle divided into slices proportional to the frequency of subgroups.

• Calculations:

  • Angle: $\text{Angle} = \frac{\text{Number of items in category}}{\text{Total number}} \times 360^\circ$

  • Percentage: $\text{Percentage} = \frac{\text{Angle}}{360^\circ} \times 100$

• Results for Commerce Students (Total $n=24$):

  • $0$ As: $30^\circ$ ($8.3\%$

  • $1$ A: $90^\circ$ ($25.0\%$

  • $2$ As: $135^\circ$ ($37.5\%$

  • $3$ As: $60^\circ$ ($16.7\%$

  • $4$ As: $45^\circ$ ($12.5\%$

• Results for Facebook Age Groups:

  • $13 - 25$: $\frac{45}{100} \times 360 = 162^\circ$

  • $26 - 44$: $\frac{36}{100} \times 360 = 129.6^\circ$

  • $45 - 64$: $\frac{19}{100} \times 360 = 68.4^\circ$