5.2 Data Displays, Mean/Median/Mode & Basic Algebraic Equations

Data Displays, Mean/Median/Mode & Basic Algebraic Equations

Data handling and algebraic reasoning are core pillars of quantitative literacy. Data handling encompasses collecting, organizing, presenting, and interpreting quantitative facts to discover patterns and make evidence-based predictions. Introductory algebra introduces symbolic notation—using letters as variables—to represent unknown numbers, model mathematical relationships, and solve multi-step real-world problems. On national primary examinations in Jamaica, students are expected to extract insights from diverse data displays, compute statistical benchmarks, evaluate likelihood on a probability scale, and solve linear equations using inverse operations.

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Data Collection, Tallying & Frequency Distributions

All statistical investigations begin with the collection of raw data. Raw data refers to unprocessed numerical or categorical facts gathered directly from surveys, observations, or experiments.

The Tally System

To organize raw data without omission or duplicate counting, statisticians utilize tally marks. Tally marks are recorded as individual vertical strokes ($|$). When four vertical strokes are made, the fifth count is drawn as a diagonal slash crossing through the four strokes, forming a distinct group or "bundle" of 5:

• A tally of 3: $|||$
• A tally of 5: Four vertical strokes with a diagonal slash
• A tally of 14: Two bundles of 5 plus four individual vertical strokes ($5 + 5 + 4 = 14$)

Bundling in groups of five facilitates rapid skip-counting ($5, 10, 15, \dots$) and minimizes counting errors when tallying large cohorts.

Frequency Tables

A frequency table synthesizes data into three organized columns:

1. Category / Data Value: The attribute being observed (e.g., preferred sport, shoe size, test score).
2. Tally Marks: The visual counting record grouped in bundles of five.
3. Frequency ($f$): The total count written as a standard Arabic numeral.

The sum of all frequencies equals the total sample size ($N$ or $\sum f$).

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Graphical Data Displays

Once organized into frequency tables, data can be communicated through visual graphs. Selecting the appropriate graphical display depends on the type of data (discrete categorical versus continuous time series) and the purpose of the comparison.

1. Pictographs (Picture Graphs)

A pictograph uses icons or picture symbols to represent quantities of data.

• The Key (Legend): Every pictograph must feature an explicit key defining what quantity each symbol represents (e.g., $\text{Key: } 1 \text{ star} = 10 \text{ students}$).
• Fractional Symbols: When a data value is not a multiple of the key, a fractional portion of the symbol is used:
  • If $1 \text{ star} = 10 \text{ students}$, then half a star represents $10 \div 2 = 5$ students.
  • If $1 \text{ circle} = 20 \text{ vehicles}$, a three-quarter circle represents $\frac{3}{4} \times 20 = 15$ vehicles.
• Exam Trap: Never count picture symbols as single units without first checking the key. Always multiply the count of full and fractional symbols by the stated key multiplier.

2. Bar Charts (Column Graphs)

A bar chart represents discrete categorical data using rectangular bars of uniform width. The length or height of each bar is proportional to the frequency it represents.

• Vertical Bar Chart: Categories appear along the horizontal axis ($x$-axis), while frequencies are measured along the vertical axis ($y$-axis).
• Horizontal Bar Chart: Frequencies run along the horizontal axis, and categories are listed on the vertical axis (useful when category names are long).
• Essential Structural Conventions:
  1. Bars must have equal widths.
  2. Spaces between bars must be uniform (spaces show that categories are discrete and distinct, unlike a continuous histogram).
  3. The frequency axis must feature a consistent, uniform scale starting from 0 (e.g., intervals of $2, 5, 10,$ or $20$).
  4. Both axes must have clear labels, and the entire graph must have a descriptive title.

3. Double Bar Charts

A double bar chart places two contrasting bars side-by-side for each category. This allows direct, simultaneous visual comparison between two related sub-groups (such as Boys versus Girls, Grade 5 versus Grade 6, or Rainfall in 2024 versus 2025).

• A distinct legend / key with contrasting colors or hatching patterns is mandatory to distinguish the two data series.
• Bar pairs within each category touch, but a clear gap separates one category pair from the next.

4. Line Graphs

A line graph displays continuous data that changes over chronological time intervals (time-series data). Points are plotted on a coordinate grid and joined by straight line segments.

• Typical Applications: Hourly temperature readings, monthly rainfall totals, height growth of a child over years, or money saved over consecutive weeks.
• Reading Trends: A rising line indicates an upward trend (increase); a falling line indicates a downward trend (decrease); a horizontal line indicates no change (constant).
• Interpolation: Estimating a value between two known plotted time points by reading across from the line segment.

5. Pie Charts (Circle Graphs)

A pie chart displays data as proportional sectors of a circular disk. The entire circular disk represents the whole population ($100\%$ or total frequency), corresponding to a complete turn of $360^\circ$.

To construct or interpret a pie chart:

1. Find the Fraction of the Whole: $\text{Fraction} = \frac{\text{Category Frequency}}{\text{Total Frequency}}$
2. Calculate the Central Angle for the Sector: $\text{Sector Angle} = \text{Fraction} \times 360^\circ = \left( \frac{\text{Category Frequency}}{\text{Total Frequency}} \right) \times 360^\circ$
3. Calculate the Category Percentage: $\text{Percentage} = \text{Fraction} \times 100\% = \left( \frac{\text{Category Frequency}}{\text{Total Frequency}} \right) \times 100\%$

To extract actual frequency from a given sector angle: $\text{Category Frequency} = \left( \frac{\text{Sector Angle}}{360^\circ} \right) \times \text{Total Frequency}$

Standard Benchmark Sector Angles:

• Right angle ($90^\circ$): $\frac{90^\circ}{360^\circ} = \frac{1}{4} = 25\%$
• Straight line / Diameter ($180^\circ$): $\frac{180^\circ}{360^\circ} = \frac{1}{2} = 50\%$
• Sector of $60^\circ$: $\frac{60^\circ}{360^\circ} = \frac{1}{6} \approx 16.7\%$
• Sector of $120^\circ$: $\frac{120^\circ}{360^\circ} = \frac{1}{3} \approx 33.3\%$

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Measures of Central Tendency & Statistical Spread

Summary statistics reduce an entire dataset to a single representative value that describes where the center of the data lies (central tendency) or how widely dispersed the values are (spread).

1. The Mean (Arithmetic Average)

The mean is the most common measure of average. It is computed by finding the sum of all values in the dataset and dividing by the total number of values:

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Total count of values}} = \frac{\sum x}{n}$

• Reversing the Mean: If the mean of $n$ numbers is known, their total sum is: $\text{Total Sum} = \text{Mean} \times n$
• Key Characteristic: The mean takes every single value into account, but it is heavily distorted by extreme values (unusually high or low outliers).

2. The Median (Middle Value)

The median is the value that occupies the exact middle position when all numbers in a dataset are sorted in numerical order from least to greatest (ranked data).

• Rule for Odd Number of Values ($n$ is odd):
  • There is exactly one central value located at position $\frac{n + 1}{2}$.
  • In the sorted set $\{4, 7, 9, 12, 18\}$ ($n = 5$), the median is the 3rd number: $9$.
• Rule for Even Number of Values ($n$ is even):
  • There are two middle values located at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.
  • The median is the arithmetic mean of these two middle numbers.
  • In the sorted set $\{14, 18, 22, 26, 30, 34\}$ ($n = 6$), the two middle values are $22$ and $26$: $\text{Median} = \frac{22 + 26}{2} = \frac{48}{2} = 24$
• Key Characteristic: The median is unaffected by extreme outliers, making it the preferred measure of center when skewed distributions or unusual extremes occur.

3. The Mode (Most Frequent Value)

The mode is the data value that appears with the greatest frequency.

• A dataset can have:
  • One mode (Unimodal): A single value appears more frequently than all others (e.g., in $\{3, 5, 5, 7, 8\}$, the mode is $5$).
  • Two modes (Bimodal): Two values tie for the highest frequency (e.g., in $\{2, 4, 4, 6, 7, 7, 9\}$, the modes are $4$ and $7$).
  • No mode: All values appear with equal frequency (e.g., in $\{12, 15, 19, 23\}$, there is no mode).
• Key Characteristic: The mode is the only measure of central tendency that can be used for non-numerical, categorical data (e.g., most popular brand of shoe, most frequent eye color).

4. The Range (Measure of Spread)

The range measures how widely spread out the numbers in a dataset are. It is calculated by subtracting the smallest data value from the largest data value:

$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

The range is a measure of dispersion, not a measure of central tendency. A small range indicates tightly clustered data, while a large range indicates high variability.

Statistical Summary Reference Table

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Probability: The Likelihood of Events

Probability is the branch of mathematics that quantifies the likelihood or chance of an event occurring. Probability values range strictly from 0 to 1 (or $0\%$ to $100\%$).

The Probability Scale

• $P = 0$ (Impossible): The event cannot occur under any circumstances (e.g., rolling an 8 on a standard 6-sided die).
• $0 < P < 0.5$ (Unlikely): The event has less than a $50\%$ chance of occurring (e.g., drawing the Ace of Spades from a 52-card deck: $P = \frac{1}{52}$).
• $P = 0.5$ (Equal / Even Chance): The event has exactly a $50\%$ chance of occurring (e.g., flipping heads on a fair coin: $P = \frac{1}{2}$).
• $0.5 < P < 1$ (Likely): The event has greater than a $50\%$ chance of occurring (e.g., rolling a number greater than 1 on a 6-sided die: $P = \frac{5}{6}$).
• $P = 1$ (Certain): The event is guaranteed to happen (e.g., picking a yellow counter from a bag containing only yellow counters).

Theoretical Probability Formula

When all possible outcomes in an experiment are equally likely, the theoretical probability of an event $E$ is calculated as:

$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$

Complementary Events

The complement of an event $E$ is the event that $E$ does not occur, written as $\text{not } E$. Because an event either happens or does not happen, their probabilities always sum to 1:

$P(E) + P(\text{not } E) = 1 \implies P(\text{not } E) = 1 - P(E)$

For example, if the probability of rain tomorrow is $\frac{3}{10}$ ($30\%$), the probability that it will not rain is $1 - \frac{3}{10} = \frac{7}{10}$ ($70\%$).

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Introductory Algebraic Thinking: Variables and Expressions

Algebra introduces symbolic thinking into arithmetic. Instead of working solely with known numbers, we use letters (called variables) as placeholders for unknown or changing quantities.

Core Terminology

• Variable: A letter or symbol representing an unknown numerical value (e.g., $x, y, n, p$).
• Constant: A fixed numerical quantity that does not change (e.g., $+7, -12$).
• Coefficient: The numerical factor multiplying a variable. In the term $5n$, the coefficient is $5$. (When no number is written before a variable, such as $x$, the coefficient is understood to be $1$).
• Term: A single number, variable, or product of numbers and variables separated by $+$ or $-$ signs (e.g., in $3x + 8$, $3x$ and $8$ are terms).

Algebraic Expressions versus Algebraic Equations

It is vital to distinguish between an expression and an equation:

• Algebraic Expression: A mathematical phrase containing numbers, variables, and operation symbols, but no equals sign (e.g., $4n + 9$). Expressions can be evaluated or simplified, but they cannot be "solved."
• Algebraic Equation: A complete mathematical statement showing that two expressions are equal, joined by an equals sign ($=$) (e.g., $4n + 9 = 33$). Equations can be solved to find the specific value of the variable that makes the statement true.

Translating Verbal Statements into Algebraic Expressions

Mathematical modeling requires converting English word descriptions into symbolic algebra:

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Solving Linear Equations Using Inverse Operations

An equation functions like a balanced two-pan balance scale:

$\text{Left-Hand Side (LHS)} = \text{Right-Hand Side (RHS)}$

The fundamental principle of algebra is: Whatever operation is performed on one side of an equation must be performed identically on the other side to maintain balance.

The Method of Inverse Operations

To find the unknown variable, we "undo" operations in reverse order using their mathematical inverses:

• The inverse of Addition is Subtraction (and vice-versa).
• The inverse of Multiplication is Division (and vice-versa).

1. One-Step Linear Equations

A one-step equation requires a single inverse operation to isolate the variable:

• Addition Equations: Undo addition by subtracting from both sides: $x + 18 = 45 \implies x = 45 - 18 \implies x = 27$
• Subtraction Equations: Undo subtraction by adding to both sides: $y - 14 = 39 \implies y = 39 + 14 \implies y = 53$
• Multiplication Equations: Undo multiplication by dividing both sides: $6k = 78 \implies k = \frac{78}{6} \implies k = 13$
• Division Equations: Undo division by multiplying both sides: $\frac{m}{4} = 15 \implies m = 15 \times 4 \implies m = 60$

2. Two-Step Linear Equations ($ax + b = c$)

Two-step equations involve two operations. To isolate the variable cleanly without fractions, we undo operations in reverse order of the standard order of operations:

1. Step 1: Eliminate the constant term by adding or subtracting it from both sides.
2. Step 2: Eliminate the coefficient by multiplying or dividing both sides.
3. Step 3: Verify your solution by substituting the answer back into the original equation.

Example: Solving $4x + 15 = 47$

1. Undo $+15$ by subtracting $15$ from both sides: $4x + 15 - 15 = 47 - 15$ $4x = 32$
2. Undo multiplication by $4$ by dividing both sides by $4$: $\frac{4x}{4} = \frac{32}{4}$ $x = 8$
3. Check: $4(8) + 15 = 32 + 15 = 47$. Both sides match; the solution $x = 8$ is correct.

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Step-by-Step Worked Problem Walkthroughs

Worked Example 1: Finding a Missing Test Score Given a Target Mean

Problem: Keisha has sat four term tests in Mathematics and received scores of $76$, $84$, $88$, and $80$. What score must she achieve on her fifth and final test to finish the term with an overall mean score of $84$?

Solution Walkthrough:

1. Determine the Total Number of Tests: There are $5$ tests in total ($n = 5$).
2. Calculate the Required Total Points: To have a mean of $84$ across $5$ tests: $\text{Required Total Sum} = \text{Target Mean} \times n = 84 \times 5 = 420 \text{ points}$
3. Calculate the Current Points Earned: Add Keisha's four completed test scores: $\text{Current Sum} = 76 + 84 + 88 + 80 = 328 \text{ points}$
4. Find the Missing Score by Subtraction: Let $s$ represent the score on the fifth test: $328 + s = 420$ $s = 420 - 328 = 92$
5. Verify: $\text{Mean} = \frac{76 + 84 + 88 + 80 + 92}{5} = \frac{420}{5} = 84$ Keisha must achieve a score of $92$ on her fifth test.

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Worked Example 2: Modeling a Real-World Scenario with a Two-Step Equation

Problem: Andre bought $6$ identical exercise books and a mathematical instrument set costing $320. If his total bill was $1,760, write an algebraic equation to represent this purchase and determine the price of one exercise book.

Solution Walkthrough:

1. Define the Variable: Let $b$ represent the cost of one exercise book in Jamaican dollars.
2. Formulate the Equation:
   • The cost of 6 exercise books is $6b$.
   • Adding the instrument set ($320) gives the total cost: $6b + 320 = 1,760$
3. Solve Using Inverse Operations:
   • Step 1: Subtract $320$ from both sides: $6b = 1,760 - 320$ $6b = 1,440$
   • Step 2: Divide both sides by $6$: $b = \frac{1,440}{6} = 240$
4. Verify: $6 \times 240 + 320 = 1,440 + 320 = 1,760$. One exercise book costs $240.

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Worked Example 3: Interpreting a Pie Chart and Sector Angles

Problem: A school survey investigated the favorite fruit of $180$ Grade 6 students. The pie chart results showed:

• Mango: Sector angle $= 120^\circ$
• Banana: Sector angle $= 90^\circ$
• Papaya: Sector angle $= 50^\circ$
• Orange: Sector angle $= x^\circ$ Find the angle $x^\circ$ representing Orange, calculate how many students chose Mango, and find what fraction of the students chose Banana.

Solution Walkthrough:

1. Find Angle $x^\circ$ for Orange: A complete circle measures $360^\circ$: $120^\circ + 90^\circ + 50^\circ + x^\circ = 360^\circ$ $260^\circ + x^\circ = 360^\circ$ $x^\circ = 360^\circ - 260^\circ = 100^\circ$
2. Calculate the Number of Students Who Chose Mango: $\text{Students} = \left( \frac{\text{Sector Angle}}{360^\circ} \right) \times \text{Total Students}$ $\text{Students} = \left( \frac{120^\circ}{360^\circ} \right) \times 180 = \frac{1}{3} \times 180 = 60 \text{ students}$
3. Find the Fraction for Banana: $\text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4}$ 60 students chose Mango, and $\frac{1}{4}$ of the students chose Banana.

Stem-and-Leaf Plots and Substitution

Reading and Building a Stem-and-Leaf Plot

A stem-and-leaf plot keeps every data value while showing the shape of the data. For two-digit numbers, the stem is the tens digit and the leaf is the ones digit. Leaves are written in order.

Scores of ten students: 23, 27, 31, 35, 35, 38, 42, 46, 49, 50.

Key: $3 \mid 5$ means $35$.

• How many scores are greater than 40? Count the leaves after stem 4 and stem 5: 42, 46, 49 and 50 make 4 scores.
• Mode: the repeated leaf 5 on stem 3 shows the mode is $35$.
• Median: with 10 values, average the 5th and 6th values: $\frac{35 + 38}{2} = 36.5$.
• Range: $50 - 23 = 27$.

Always read the key: if the key said $3 \mid 5 = 3.5$, the same plot would show decimal values.

Substituting into Formulae and Number Sentences

Substitution means replacing each letter with its given value and then following the order of operations.

• Perimeter of a rectangle: $P = 2(l + w)$. If $l = 9$ and $w = 4$, then $P = 2(9 + 4) = 2 \times 13 = 26$.
• If $a = 3$ and $b = 5$: $4a + 2b = 12 + 10 = 22$ and $ab - a = 15 - 3 = 12$. Remember that $ab$ means $a \times b$.
• Completing number sentences with operation symbols: $24 \;\square\; 6 = 4$ needs $\div$; $7 \;\square\; 8 = 56$ needs $\times$; $45 \;\square\; 17 = 28$ needs $-$.
• Simple inequalities: for whole numbers, $3n + 2 < 20$ is true when $3n < 18$, so $n$ can be $0, 1, 2, 3, 4$ or $5$.