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Data Interpretation

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Differential Equation
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- One-Sided Limits
- Optimization Problems
- P Series
- Particular Solutions to Differential Equations
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal´s Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation Math
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Data interpretation refers to the process of subjecting data to predefined processes such as the organization of tables, charts, or graphs so that logical and statistical conclusions can be derived. This part of statistics answers a common question among researchers: what exactly are we supposed to present?

It is not ideal for researchers to present numerical values of data collected from instruments or surveys. Data need to be organized to tell the story of what you want to emphasize in your research. This should focus on the problem you want to solve - also known as 'the statement of the problem'. It is the primary function of the research.

Statistical tools are used in the process, helping you to transform data into useful information that can help you to arrive at important conclusions. This process is called data analysis. It is after this process that data can be fully interpreted.

Statistical methods allow you to work on your data. Imagine you have the exam scores for 100 students, and you want to interpret this data. Scanning through the scores by eye alone might be quite tough! Here are two methods that would simplify this task.

Central tendency values are used to describe some key characteristics of the whole data set, producing a single value that is typical of the whole set. For example, the mode will give you the value that occurs the most often.

The

**mean**is the most commonly reported measure of central tendency and it is the mathematical average. To calculate your mean, you add up all of your values available and divide that by the number of values you added. The mean is represented by μ, and its formula is, where n is the number of data items in the sample and is the sum of all data values.

The

**median**is the mid-point value in your data set. Where the median is two numbers, it is the average of both values in ordered data.

The

**mode**is the value that occurs the most often.

Another statistical measure that is commonly used is variability, also known as spread. The range is the simplest form of variability. Let's take the exam score dataset again - the range is the span between the lowest and highest numerical values.

Another common measure is variance; which is the squared average deviation from the mean. This number indicates how much the individual values deviate from the mean. What you will see reported more often is the standard deviation. This is modelled as the square root of the variance. Standard deviation expresses how much individual class scores differ from the mean value for the group. Mathematically, it can be modelled into an equation:

Single variable data involves examining one particular variable relevant to a dataset. Single data analysis is common in descriptive forms of analysis and uses histograms, frequency distributions, and box plots among other methods. This is mostly used in the first step of investigating data. Let's take a look at a box plot.

A box plot displays a five-number summary of a dataset. They are the minimum, first quartile, median, third quartile, and maximum. Quartiles tell us about the spread of data by breaking the data set into quarters. The lower quartile, Q_{1} represents 25%, the middle quartile equals 50% and the upper quartile represents 75%.

The ages of 10 students in grade 12 were collected and they are as follows:

15, 21, 19, 19, 17, 16, 17, 18, 19, 18.

Let's first arrange these in ascending order.

15, 16, 17, 17, 18, 18, 19, 19, 19, 21.

We can now find the median, which is the middle number. And since we have an even number, we have two of them. Finding the average is standard practice; however, with ours, we have the same number.

median = 18

We will find the quartiles now. The first is the median to the left of the overall median.

That will mean we are finding the median for 15, 16, 17, 17, 18.

This equals 17.

The third quartile will be the median to the right of the median.

18, 19, 19, 19, 21

Which will make that 19.

Now we will document the minimum number which is 15.

And also document the maximum which is 21.

The image above is the box plot representing the data of the ages of the students in grade 12.

We will take another example with an odd number of data points.

The table below is data of basketball players' points scored per game over a seven-game span. Visualise this on a box and whisker plot.

Game | Points |

1 | 10 |

2 | 17 |

3 | 5 |

4 | 32 |

5 | 16 |

6 | 18 |

7 | 20 |

Step 1.

Rearrange the values in the data set from lowest to highest.

5, 10, 16, 17, 18, 20, 32.

Step 2.

Now identify the highest and lowest values in the data set

Highest value: 32

Lowest value: 5

Step 3.

We can now identify the midpoint value (median) of the data set.

Median = 17

Step 4.

We will now find the upper and lower quartiles.

The lower quartile is the median for the first half of the data set.

That will mean that we are finding the median for 5, 10, 16

Lower quartile = 10

The upper quartile is the median for the second half of the data set.

That will also mean that we are finding the median for 18, 20, 32

Upper quartile = 20

Step 5.

Now that we have all our necessary values, we will construct our box and whisker plot.

Highest value = 32

Lowest value = 5

Median = 17

Upper quartile = 20

Lower quartile = 10

We will first draw a number line that fits the data, and plot all the necessary values we found.

Construct a rectangle that encloses the median of the entire data set that its vertical lines pass through the upper and lower quartiles. Now construct a vertical line through the median that hits both ends of the rectangle.

There, we have our box and whisker plot for the basketball games.

In contrast to single variable data, bivariate data consist of two variables for each individual. For example, in large studies in the health sector, it is common to collect variables such as height, age, blood pressure, etc. in each individual. Let's look at an example in a two-way frequency table.

These are the number of males and females who had each grade on a math project in school.

degrees | Female | Total | |

A | 9 | 21 | |

B. | 18 | 32 | |

C | 11 | ||

D | 2 | 3 | 5 |

E | 1 | 2 | 3 |

Total | 38 | 42 | 80 |

We can see there are 9 males and 12 females who got an A, 18 males and 14 females who got a B, and so on.

Now we can answer a couple of questions.

How many students in total had an A?

Answer: 21 students.

How many males were surveyed?

Answer: 38 males.

How many males earned a grade of A?

Answer: 9.

Below is a graph representation of two variables, the sales of ice cream in a given shop against the temperature of the day. This demonstrates how much ice cream is purchased at every given temperature.

Probability is the measure of how likely an event is to happen. Probabilities can be placed on a number line between 0 and 1, as shown below.

So if the probability of an event is zero, then it is impossible for the event to occur. Whilst if it is 1, then it is certain. Then we have variant degrees in between the two values, and 0.5 would mean there is an even chance of the event happening.

Probabilities are written down using the following notation :

P (A): the probability of event A happening.

P (A '): the probability of event A not happening.

If event A has a between happening and not happening, then the probability of event A not happening = 1 - P (A ')

For example, if the P (A) = 0.8

P(A') = 0.2.

They should both add up to 1.

These are the basic concepts you would be using throughout probability at this level. You can be reintroduced to Venn diagrams, tree diagrams, etc. as well!

- Data interpretation refers to the process of subjecting collected data to predefined processes so logical and statistical conclusions can be derived.
- Presentation refers to the representation of data in graphs, plots, frequency tables, etc.
- The measure of central tendency produces a single value that is typical of the whole set. The basic values are mean, mode and median.
- Single variable data involves examining one particular variable relevant in a dataset.
- In contrast to single variable data, bivariate data consist of two variables for each individual.
- Probability is the measure of how likely an event is to happen.

You carry out analysis by selecting each component of the data and seeing if there are any patterns.

Data interpretation involves explaining what these findings mean with reference to the statement of the problem.

It's necessary to organise and group ideas in a logical way.

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