Types of Data and Summary Statistics Explained

📘 Topic 1: Types of Data & Summary Statistics

Level: IB SL/HL Mathematics (AA & AI), A-Level Mathematics, Further Mathematics
Tags: Descriptive Statistics, Data Exploration, Data Types, Measures of Central Tendency, Measures of Dispersion, Statistical Representation, Data Interpretation

🎯 Learning Objectives
Upon completing this topic, you will be able to:

Distinguish between different types of data (qualitative/categorical and quantitative; nominal, ordinal, discrete, continuous) and understand the implications of these distinctions for analysis.
Select and construct appropriate graphical representations for different types of data (bar charts, histograms, box plots, cumulative frequency graphs, dot plots, stem-and-leaf plots).
Calculate and interpret measures of central tendency (mean, median, mode) for various datasets.
Calculate and interpret measures of dispersion (range, interquartile range, variance, standard deviation) for various datasets.
Understand the properties of these summary statistics, including their sensitivity to outliers and their relevance to data distribution shape.
Utilize a GDC/calculator efficiently for statistical calculations.
Critically interpret and compare statistical representations and summary statistics to draw meaningful conclusions about data.

🌟 Introduction: Why Study Descriptive Statistics?
Descriptive statistics are the foundational tools for understanding any dataset. Before we can make complex inferences or predictions, we need to summarize and describe the main features of the data we have. This involves identifying patterns, variability, central values, and potential anomalies. Mastering these concepts allows you to effectively communicate insights from data and provides the groundwork for more advanced statistical methods.

1️⃣ Types of Data: The Building Blocks

Understanding the type of data you’re working with is the first crucial step, as it dictates the types of statistical analyses and graphical representations that are appropriate.

Category	Subtypes	Definition	Examples	Key Characteristics
Qualitative (Categorical)	Data that describes qualities or characteristics. Cannot be arithmetically manipulated in a meaningful way (e.g., you can’t average eye colors).
	Nominal	Categories with no intrinsic order or ranking.	Eye color (blue, brown, green), Gender (male, female, other), Car Make (Ford, Toyota, BMW), Postcodes	Labels or names. Operations like counting frequency per category are common.
	Ordinal	Categories with a meaningful order or ranking, but the differences between categories are not necessarily equal or quantifiable.	Satisfaction rating (Very Dissatisfied, Dissatisfied, Neutral, Satisfied, Very Satisfied), Education Level (High School, Bachelor’s, Master’s, PhD), T-shirt size (S, M, L, XL)	Order matters. Can determine if one category is “more” or “less” than another, but not “how much more/less.”
Quantitative (Numerical)	Data that represents amounts or counts. Can be arithmetically manipulated.
	Discrete	Data that can only take specific, separate numerical values (often integers). Usually arises from counting.	Number of students in a class, Number of cars in a parking lot, Shoe size (e.g., 7, 7.5, 8 – though shoe sizes can be tricky and sometimes treated as ordinal or even continuous depending on context), Number of defective items.	Countable. There are “gaps” between possible values (e.g., you can’t have 2.5 students).
	Continuous	Data that can take any value within a given range. Usually arises from measuring.	Height (e.g., 175.32 cm), Weight (e.g., 68.7 kg), Temperature (e.g., 25.5 °C), Time taken to complete a race (e.g., 10.34 seconds).	Measurable. Can be infinitely divided into finer and finer values (limited only by measuring instrument precision).

📌 Note:

The distinction between data types is crucial. For instance, calculating the “mean” of nominal data like eye color is meaningless.
Sometimes, data can be borderline. For example, age is technically continuous, but we often treat it as discrete (e.g., “18 years old”). Shoe sizes can be discrete (e.g. 7, 7.5, 8) but reflect an underlying continuous foot length. Context is key.

2️⃣ Representing Data: Visualizing Information

Visualizations help us understand patterns, trends, and outliers in data more intuitively.

Graph Type	Best For	Key Features & Interpretation Insights
Bar Charts	Categorical data (Nominal or Ordinal). Comparing frequencies or proportions across categories.	Bars are of equal width and separated by gaps. Height of bar represents frequency or relative frequency. Easy to compare categories.
Histograms	Grouped continuous data (or discrete data with many values). Showing the distribution (shape, center, spread) of the data.	Bars touch (unless a class interval has zero frequency). X-axis represents continuous scale divided into class intervals. Area of bar proportional to frequency. Shape (symmetrical, skewed), modality (uni-, bi-modal), gaps, outliers.
Box Plots (Box and Whisker Plots)	Displaying the five-number summary (Min, Q1, Median, Q3, Max) and identifying potential outliers. Excellent for comparing distributions across multiple groups.	Shows median (Q2), interquartile range (IQR = Q3-Q1) as the box, and range. Whiskers extend to min/max values within a certain range (e.g., 1.5 x IQR from the box). Points beyond whiskers are outliers. Good for assessing symmetry and spread.
Cumulative Frequency Graphs (Ogives)	Showing the total frequency of data points that fall below a certain value. Estimating median, quartiles, and percentiles.	S-shaped curve. Y-axis goes up to total frequency. Median is at 50% of total frequency, Q1 at 25%, Q3 at 75%. Steepness indicates density of data.
Dot Plots	Small datasets of discrete or continuous data. Showing individual data points and their frequency.	Each dot represents a single data point. Useful for seeing clusters, gaps, and distribution shape quickly for small n.
Stem-and-Leaf Plots	Small to moderate datasets of quantitative data. Preserves individual data values while showing distribution shape.	“Stem” represents leading digit(s), “leaf” represents trailing digit. Provides a quick visual of distribution, similar to a sideways histogram, but retains original data values.
Frequency Tables	Organizing raw data (categorical or quantitative) by showing the frequency of each value or group of values.	Can include columns for relative frequency (proportion) and cumulative frequency. Precursor to many graphs like bar charts and histograms.

⚠️ Key Distinction: Bar Chart vs. Histogram

Bar Chart: Used for categorical data. Bars are separate. X-axis has distinct categories.
Histogram: Used for quantitative (usually continuous) data. Bars touch (representing a continuous scale). X-axis is a numerical scale divided into intervals.

3️⃣ Measures of Central Tendency: Finding the “Center”

These statistics describe a typical or central value of a dataset.

Measure	Symbol(s)	Formula / Method	When to Use / Properties	Sensitivity to Outliers
Mean (Arithmetic Average)	μ (mu) for population<br> x̄ (x-bar) for sample	x̄ = (Σxᵢ) / n <br> μ = (Σxᵢ) / N <br> (Sum of all values divided by the number of values)	For quantitative data. Best for symmetrical distributions without significant outliers. Uses all data values. Often preferred for further statistical inference.	High
Median	M, Med, Q2	Middle value when data is ordered. <br> If n is odd: (n+1)/2 <sup>th</sup> value. <br> If n is even: average of n/2 <sup>th</sup> and (n/2 + 1)<sup>th</sup> values.	For quantitative data (and sometimes ordinal). Best for skewed distributions or when outliers are present. Not affected by extreme values.	Low (Robust)
Mode	–	Most frequent value(s) in the dataset.	For all types of data (qualitative and quantitative). Can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal). Useful for categorical data.	Low

Population (N): All members of a defined group.
Sample (n): A subset of a population.

4️⃣ Measures of Dispersion (Spread or Variability): How Spread Out is the Data?

These statistics describe how much the data points vary from each other or from the center.

Measure	Formula / Definition	Interpretation & Properties	Sensitivity to Outliers
Range	Max Value – Min Value	Simplest measure of spread. Highly influenced by extreme values. Provides a quick, but often crude, indication of variability.	Very High
Interquartile Range (IQR)	IQR = Q3 – Q1 <br> (Q3: Upper Quartile, 75th percentile; Q1: Lower Quartile, 25th percentile)	Measures the spread of the middle 50% of the data. Robust to outliers. Often used with the median. A smaller IQR indicates less variability in the central portion of the data. Useful for constructing box plots and identifying potential outliers (values outside Q1 – 1.5IQR or Q3 + 1.5IQR).	Low (Robust)
Variance	Population (σ²): σ² = Σ(xᵢ – μ)² / N <br> Sample (s²): s² = Σ(xᵢ – x̄)² / (n–1)	Average of the squared deviations from the mean. Units are squared (e.g., cm² if data is in cm), making interpretation less direct. The (n-1) denominator for sample variance provides an unbiased estimate of the population variance.	High
Standard Deviation	Population (σ): σ = √[Σ(xᵢ – μ)² / N] <br> Sample (s): s = √[Σ(xᵢ – x̄)² / (n–1)]	Square root of the variance. Measures the typical or average deviation of data points from the mean. Expressed in the same units as the original data, making it more interpretable than variance. A small SD means data points tend to be close to the mean; a large SD means data points are spread out.	High

📌 Population (σ, σ²) vs. Sample (s, s²):

Use population formulas if your dataset includes every member of the group you’re interested in.
Use sample formulas if your dataset is a subset of a larger population, and you want to estimate the population’s spread. The (n-1) in the denominator for sample variance/SD is Bessel’s correction, which provides a better (unbiased) estimate of the population parameter. Calculators usually provide both.

🤔 Choosing the Right Measures

Symmetrical Data: Mean and Standard Deviation are often preferred.
Skewed Data or Data with Outliers: Median and IQR are generally more appropriate as they are less affected by extreme values.
Relationship between Mean, Median, and Mode for Skewness:
- Symmetrical: Mean ≈ Median ≈ Mode
- Positively Skewed (skewed right): Mode < Median < Mean (tail to the right)
- Negatively Skewed (skewed left): Mean < Median < Mode (tail to the left)

5️⃣ GDC/Calculator Tips (IB – e.g., TI-84 Plus, Casio fx-CG series)

Most GDCs have built-in functions for descriptive statistics:

Enter Data:
- TI-84: Press STAT, select 1:Edit…. Enter data into a list (e.g., L1). If you have frequencies, enter them into a second list (e.g., L2).
- Casio: Go to MENU, select STAT. Enter data into a list (e.g., List 1). If frequencies, set up in SET (F6) for 1-Var Freq.
Calculate 1-Variable Statistics:
- TI-84: Press STAT, go to CALC menu, select 1:1-Var Stats.
  - If data in L1: List: L1, FreqList: (leave blank or enter 1 if no separate frequency list, or enter L2 if frequencies are in L2). Calculate.
- Casio: After entering data, press CALC (F2), then 1VAR (F1).
Output Interpretation:
- x̄: Sample mean
- Σx: Sum of data values
- Σx²: Sum of squared data values
- Sx: Sample standard deviation (uses n-1 denominator) – This is usually what you need for samples.
- σx (or σₓ): Population standard deviation (uses N denominator) – Use if your data is the entire population.
- n: Number of data points
- minX: Minimum value
- Q1Med: Lower Quartile
- Med (or Median): Median
- Q3Med: Upper Quartile
- maxX: Maximum value
- (Some calculators might also show Mode)

✅ SOLVED EXAMPLE 1: Analyzing Student Test Scores

A small group of 5 students (considered the entire population for this analysis) scored the following marks on a test: 72, 75, 78, 85, 90.

Find:
a) Mean
b) Median
c) Range
d) Interquartile Range (IQR)
e) Population Standard Deviation (σ)

Solution:
Data: {72, 75, 78, 85, 90}. Ordered: 72, 75, 78, 85, 90. (N=5)

a) Mean (μ):
μ = (72 + 75 + 78 + 85 + 90) / 5 = 400 / 5 = 80

b) Median:
Since N=5 (odd), the median is the (5+1)/2 = 3rd value.
Median = 78

c) Range:
Range = Max Value – Min Value = 90 – 72 = 18

d) Interquartile Range (IQR):

Q1: The median of the lower half. Lower half (excluding overall median if N is odd): {72, 75}.
Q1 = (72 + 75) / 2 = 147 / 2 = 73.5
(Note: Different methods for Q1/Q3 exist. GDCs typically use an inclusive/exclusive method. For IB/A-Level, be consistent with your GDC or stated method. For N=5, Q1 is often taken as the (N+1)/4 = 1.5th value, interpolating between 1st and 2nd. Or, for small N, Q1 is the median of values below the overall median.)
Using the common GDC method (TI method): (N+1)/4 = (5+1)/4 = 1.5th value. So, average of 1st and 2nd value = (72+75)/2 = 73.5
Q3: The median of the upper half. Upper half (excluding overall median if N is odd): {85, 90}.
Q3 = (85 + 90) / 2 = 175 / 2 = 87.5
Using the common GDC method (TI method): 3(N+1)/4 = 3(5+1)/4 = 4.5th value. So, average of 4th and 5th value = (85+90)/2 = 87.5
IQR = Q3 – Q1 = 87.5 – 73.5 = 14

e) Population Standard Deviation (σ):

Calculate deviations from the mean (μ=80):
(72-80) = -8
(75-80) = -5
(78-80) = -2
(85-80) = 5
(90-80) = 10
Square the deviations:
(-8)² = 64
(-5)² = 25
(-2)² = 4
(5)² = 25
(10)² = 100
Sum the squared deviations: Σ(xᵢ – μ)² = 64 + 25 + 4 + 25 + 100 = 218
Calculate Variance (σ²): σ² = Σ(xᵢ – μ)² / N = 218 / 5 = 43.6
Calculate Standard Deviation (σ): σ = √43.6 ≈ 6.60

(If this were a sample, s would be √(218 / (5-1)) = √(218/4) = √54.5 ≈ 7.38)

Using GDC (e.g., TI-84):
Enter 72, 75, 78, 85, 90 into L1.
STAT -> CALC -> 1-Var Stats (List: L1, FreqList: blank)
Output would show: x̄ = 80 (interpreted as μ here), σx = 6.603029…, Q1Med = 73.5, Med = 78, Q3Med = 87.5.

🧠 COMMON PITFALLS & THINGS TO WATCH OUT FOR

Discrete vs. Continuous Misidentification: Choosing the wrong graph (e.g., bar chart for continuous data when a histogram is needed) or statistical test later on.
Population (σ) vs. Sample (s) Standard Deviation: Using the wrong formula or GDC output. If in doubt and the data is a subset of a larger group, use sample SD (s). If the data represents everyone or everything you are concerned with, use population SD (σ).
Not Ordering Data: Forgetting to order the dataset before finding the median, quartiles, or IQR.
Bar Chart vs. Histogram Confusion: Remember: gaps for categorical (bar chart), no gaps for continuous (histogram).
Interpreting Skewness: Don’t just calculate mean and median; compare them. If mean > median, suspect positive skew. If mean < median, suspect negative skew.
Outlier Influence: Being unaware of how outliers can drastically affect the mean and standard deviation (but not so much the median and IQR).
Units of Variance: Remembering that variance is in squared units (e.g., cm²), while standard deviation is in original units (cm).
Calculating Q1/Q3: Different methods exist (inclusive/exclusive of median, interpolation formulas). GDCs have specific algorithms. For exams, usually the GDC’s method is accepted, or a method will be specified. Consistency is key.

🧪 PRACTICE QUESTIONS

🟡 Easy: Data Type Identification
Identify whether the following are qualitative (nominal/ordinal) or quantitative (discrete/continuous):

The time it takes for students to travel to school. (Quantitative Continuous)
The brand of mobile phone owned by students. (Qualitative Nominal)
The number of siblings each student has. (Quantitative Discrete)
Students’ ranking of school lunch quality on a scale of “Poor,” “Average,” “Good,” “Excellent.” (Qualitative Ordinal)
The nationalities of players in a football team. (Qualitative Nominal)

🟠 Medium: Calculations & Interpretation
A dataset representing the number of hours 10 students spent studying for an exam is:
{8, 5, 12, 3, 7, 10, 8, 6, 9, 0}

a) Calculate the mean, median, and mode.
b) Calculate the range, IQR, and sample standard deviation (s).
c) A student who studied for 0 hours is an outlier. If this student is removed, how would you expect the mean and standard deviation to change? How would the median and IQR change? (Explain, don’t recalculate fully unless you want to verify).

Solution Outline for Medium:

Ordered data: {0, 3, 5, 6, 7, 8, 8, 9, 10, 12} (n=10)
a) Mean: (0+3+…+12)/10 = 68/10 = 6.8 hours
Median: Average of 5th & 6th values = (7+8)/2 = 7.5 hours
Mode: 8 hours
b) Range: 12 – 0 = 12 hours
Q1: (10+1)/4 = 2.75th value. Using GDC method (or common interpolation/rounding), Q1 is the 3rd value = 5.
Q3: 3(10+1)/4 = 8.25th value. Q3 is the 8th value = 9.
IQR: 9 – 5 = 4 hours.
Sample SD (s): Use GDC or formula: s ≈ 3.55 hours.
c) Removing 0 (outlier):
Mean: Expected to increase (as a low value is removed).
Standard Deviation: Expected to decrease (as the data becomes less spread out).
Median: Might change slightly or not at all, less affected. (New median would be 8).
IQR: Might change slightly or not at all, less affected. (New Q1=6, New Q3=9.5, New IQR=3.5).

🔴 Hard (IB/A-Level Style Application)
A survey was conducted on 200 randomly selected adults to find the number of hours they spend on social media per week. The results are summarized in the following cumulative frequency graph.

(Imagine a typical S-shaped cumulative frequency graph where the x-axis is “Hours per week” (e.g., from 0 to 30 hours) and the y-axis is “Cumulative Frequency” (from 0 to 200). Key points might be approximately: (5 hours, 40 people), (10 hours, 100 people – median), (15 hours, 160 people), (20 hours, 190 people), (25 hours, 200 people).)

Estimate from the graph:
a) The median number of hours spent on social media.
b) The lower quartile (Q1) and upper quartile (Q3).
c) The interquartile range (IQR).
d) The number of adults who spend more than 15 hours per week on social media.
A box plot is to be drawn for this data. Calculate the values for the whiskers, assuming outliers are values more than 1.5 x IQR below Q1 or above Q3. Are there any outliers based on this definition and your estimated values? (Assume Min=0 and Max=25 for this check based on the graph’s extent).
If the mean number of hours was calculated to be 11.5 hours, what does this suggest about the skewness of the distribution when compared to your estimated median? Explain your reasoning.

Solution Approach for Hard:

a) Median: Find 0.50 * 200 = 100th person on y-axis, read across to curve, then down to x-axis. (e.g., ≈10 hours)
b) Q1: Find 0.25 * 200 = 50th person. (e.g., ≈6 hours)
Q3: Find 0.75 * 200 = 150th person. (e.g., ≈14 hours)
c) IQR = Q3 – Q1. (e.g., 14 – 6 = 8 hours)
d) Total – (Cumulative Freq at 15 hours). (e.g., 200 – 160 = 40 people)
Lower whisker boundary = Q1 – 1.5IQR. Upper whisker boundary = Q3 + 1.5IQR.
Compare estimated Min (0) and Max (25) to these boundaries. (e.g., Lower: 6 – 1.58 = 6 – 12 = -6. So, lower whisker extends to actual Min, 0. Upper: 14 + 1.58 = 14 + 12 = 26. So, upper whisker extends to actual Max, 25. No outliers in this example based on these estimations.)
Compare mean (11.5 hrs) to median (e.g., 10 hrs). Since Mean > Median, this suggests a slight positive skew (skewed to the right). This implies that while many people spend around 10 hours or less, a few spend significantly more, pulling the mean higher.

🔑 Key Takeaways

The type of data dictates your analysis.
Visualizations are essential for initial understanding.
Measures of central tendency and dispersion provide numerical summaries of key data features.
Mean/SD are sensitive to outliers; Median/IQR are robust.
Understanding population vs. sample statistics is crucial for accurate interpretation and inference.
GDCs are powerful tools, but understanding the concepts behind the calculations is paramount.

📚 Further Study & Connections

Probability Distributions: Understanding data distributions (e.g., Normal, Binomial) builds upon these descriptive foundations.
Hypothesis Testing: Comparing means or proportions often uses sample statistics (x̄, s) to make inferences about populations.
Correlation and Regression: Examining relationships between two quantitative variables.

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