Data Organization and Presentation Notes

Data Organization and Presentation (Descriptive Stats – Grouped)

Describing Data with Tables and Graphical Presentations

Intended Learning Outcomes:

• Organize data into frequency distribution, contingency tables, and appropriate graphical representation methods.

• Organize a given dataset using software application functions into appropriate graphical data presentations.

• Summarize the key findings from a set of organized data.

Scenario

Consider organizing a music festival in Singapore. To understand the target audience, look at variables like:

• Music genre

• Age group

• Preferred language

• Gender

• Household income

• Budget set aside for concerts

Raw data is chaotic until organized into readable form where meaningful conclusions can be made. Contingency tables or frequency distribution tables can be used to help organize survey data.

Contingency Table

A summary table that presents data showing the relationship between 2 or more qualitative (categorical) variables.

• Example: Finding out how many respondents like a specific music genre in a particular language.

• The number in each cell shows the frequency counts of music-lovers who enjoy a particular combination of music genre in a certain language.

Frequency Distribution Table

Groups data into categories/classes, showing the number of quantitative observations in each category/class.

• Example: Determining how much your target audience would be willing to pay for a concert ticket.

• The lowest amount in the survey was $50$ and the highest amount was $590$.

• The data collected is then organized into a frequency distribution table which summarizes the number of respondents who would be willing to pay a specific budget for performances.

Analysis

Frequency distribution table provides a simplified view of the data. Most respondents (observations) would be willing to pay between $100$ and $299$ for a concert ticket.

Further analysis helps understand:

• Range or spread of the data.

• Concentration of the data.

• Shape of the distribution (Skewness).

Constructing a Frequency Distribution Table

Step 1: Data Collection

• Collect the necessary data.

Step 2: Decide the Number of Classes

• Classes are the categories or "bins" into which you will sort your data.

• Keep between 5 to 20 classes for meaningful insights.

Step 3: Determine the Class Interval Width

• Formula: $\frac{(Highest\ value - Lowest\ value)}{Number\ of\ Classes}$

• Use intervals that are easy to understand and visualize, e.g., multiples of 5, 10, etc.

Step 4: Determine Class Limits

• Starting with the lowest value, add the Class Interval Width to get the upper limit of the first class.

• For the next class, the lower limit is the upper limit of the previous class. Continue until all classes are created.

Step 5: Tally Your Data and Find the Frequencies

• Count the frequency (occurrences) for each class.

• Check the total tally to ensure it matches the total number of observations.

Example

Survey of 100 classmates on the number of texts they sent in a day.

• Lowest: 10 texts

• Highest: 88 texts

• Number of Classes: 8

• Class Interval Width: $\frac{78}{8} = 9.75 \approx 10$

• Class Limits: Start at 10 and create ranges up to 90 (10 up to 20, 20 up to 30, and so on).

Relative Frequency Distribution and Percentage

Relative frequency gives context by using the proportions of each class in terms of the whole dataset.

• Relative Frequency = $\frac{f}{\Sigma f}$

• Relative Frequency Percentage = Relative Frequency × 100%

Example

429 respondents are willing to pay between $100$ and $200$ for a concert ticket.

Observations

• The range of amount spent per meal for Class A is bigger ($5$ to less than $45$) compared to Class B ($5$ to less than $40$).

• The modal amount spent for Class A is $15$ to less than $20$, whereas the modal class is $5$ to less than $10$ in Class B.

• For both classes, about 70% of students spend between $5$ to less than $25$ per meal.

• The shape of the distribution for both classes is skewed to the right.

Frequency Distribution and Computing Grouped Mean

Formula to compute the grouped mean:

• Grouped Mean = $\frac{\Sigma fx}{n}$

• $\Sigma fx$ refers to total weighted sum.

• $n$ refers to the total frequency (total number of observations).

Example

A grouped mean of $262.29$ means that, on average, the respondents in this survey are willing to spend about $262$ for a concert ticket.

Note

The computation of the grouped mean is only an estimation as we assume that the data points are uniformly distributed within each class which may not be true. The grouped mean does not consider the exact data points as it uses class midpoints to get the average.

Graphical Presentation of Numerical Variables

Graphical presentations such as shapes, charts, and graphs are used to communicate data in an accessible and visually appealing way. They allow us to quickly identify relationships between variables and outliers in the dataset.

Histogram

• Good for quantitative continuous data (e.g., height, weight, exam scores).

• Bars in a histogram are placed next to each other, with no gaps.

• Helps to give a sense of the distribution and shape of the data.

Bar Graph

• Good for comparing categorical data.

• Bars do not touch, representing distinct groups or categories (e.g., music genres, types of movies).

Example: Histogram

Based on the histogram, we can draw the following observations:

• Range: The range of the data is from $0$ up to $600$.

• Clustering of the data: The majority of the observations fall into 2 modal classes, audience with budgets of $100$ up to $200$ and budgets of $200$ up to $300$.

• Shape of distribution: The histogram is skewed to the right or positively skewed.

Other Types of Graphs used to present Numerical Variables

Dot Plots

These are similar to histograms but instead of bars, individual dots are used to show the distribution of data across the variables. This can be helpful for small datasets and are able to show the individual data points.

• The dot plot tells us that the modal time of visit to the store is at 1700 hours (or 5 pm), the shop is the busiest at this time.

Stem-and-Leaf Display

The stem-and-leaf display splits each data point into a “stem” and “leaf”. The “stem” is the first digit, whilst the “leaf” is the last digit. It is a simple way of showing how the data is distributed whilst retaining the value of the data point.

• The stem-and-leaf display splits each data point into a “stem” and “leaf”.

• The “stem” is the first digit, whilst the “leaf” is the last digit.

• It is a simple way of showing how the data is distributed whilst retaining the value of the data point.

Example

• The "1" stem represents ages in the teens (10-19).

• The "2" stem represents ages in the twenties (20-29).

• The "3" stem represents ages in the thirties (30-39).

• This display tells us the frequency of each age and the distribution across different age groups.

Cumulative Frequency Graphs

The cumulative frequency graph shows us the accumulation of frequencies at or below each class which can be helpful to understand the distribution and relative frequencies.

• The curve tells us that about 26 packets of cookies weigh 80 grams or less, and there are no packets of cookies that weigh more than 200 grams.

Summary

Frequency distribution tables organize our data into classes, making it easier to see how often specific values occur, which is essential for identifying the most common or rare occurrences within a dataset. Graphs, such as histograms help to provide a visual representation of the data's distribution. Together, these descriptive statistics tools provide a clear, concise way to organise and present data, and communicate findings to both technical and non-technical audiences.