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Statistical Measures

- 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
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
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- 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
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- 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
- Integrating Even and Odd Functions
- Integration Tables
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- Jump Discontinuity
- Limit Laws
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- Linear Differential Equation
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- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
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- Net Change Theorem
- Newton's Method
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- P Series
- Particular Solutions to Differential Equations
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- Decision Maths
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- Quartiles and Interquartile Range
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- Algebra
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- Arithmetic Sequences
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- Interest
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- Iterative Methods
- Law of Cosines in Algebra
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- Laws of Logs
- Limits of Accuracy
- Linear Expressions
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- Location of Roots
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- Math formula
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- Notation
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- Representation of Complex Numbers
- Rewriting Formulas and Equations
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- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
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- Sequences
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- Similar Triangles
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- Simultaneous Equations
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- Types of Functions
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- Verifying Trigonometric Identities
- Writing Equations
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- Statistics
- Binomial Distribution
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- Bivariate Data
- Box Plots
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- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- 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

**Statistical measures **are a descriptive analysis technique used to summarise the characteristics of a data set. This data set can represent the whole population or a sample of it. Statistical measures can be classified as measures of central tendency and measures of spread.

**Measures of Central Tendency **describe some key characteristics of the data set based on the average or middle values, as they describe the centre of the data. The measures of central tendency that we will be looking at are the mean, mode, and median.

The **mean**, also called the mathematical average of a given data set, can be found by adding all values in the data set, and dividing by the number of values. We can use a mathematical formula to describe this : where 𝜇 is used to represent the mean.

We have the scores of a quiz taken by mathematics students in the grade. They are 76, 89, 45, 50, 88, 67, 75, 83. What is the mean score?

Answer:

The formula above means we will add all the scores and then divide the sum by the number of scores available.

76 + 89 + 45 + 50 + 88 + 67 + 75 + 83 = 573

Since there are 8 scores available, we will divide our sum by 8.

𝜇 = 71.625

The **mode **is the most frequently occurring value in a data set. Sometimes you will have a data set where this describes more than one value. Here they are all considered the mode.

Find the mode for the given data set 6, 9, 3, 6, 6, 5, 2, 3.

Answer:

Arranging these values in ascending order will help you to identify which one occurs the most.

2, 3, 3, 5, 6, 6, 6, 9

It is evident that 6 is the most frequently occurring number, therefore the mode is 6.

The **median **is the midpoint value of a given data set. In cases where the midpoint values are two (when the number of data points is even), you need to find the average of both middle values. When finding the median, it is appropriate to reorder your values in ascending order. Take the value if the number of data points is odd. When the number is even, take the and the** value.**** **

The ages of 12 students in grade 11 were collected, and the values are as follows: 15, 21, 19, 19, 20, 18, 17, 16, 17, 18, 19, 18. Find the median age.

Answer:

Arrange these values in ascending order:

15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 21

Since the number of data points is even, we will have two middle numbers, which are both 18. So the median is 18.

The scores of an exam taken by 7 students are given below. Find the median score.

87, 56, 78, 66, 73, 71, 79

Answer:

Rearrange the numbers from lowest to highest.

56, 66, 71, 73, 78, 79, 87

The number of value points is odd, so the middle number becomes the median score.

Median = 73

**Measures of spread **are statistical measures that describe the similarity and variety of the values of given datasets. Relying on central tendency measures alone as a summary description for data sets can be very misleading since it does not account for extreme values. Measures of spread help us do that, including range, variance, and standard deviation.

**The range **is the difference between a given data set's highest and lowest values. It helps you to know how wide the data is. To find the range, the lowest value in the data is subtracted from the highest value.

Find the range of the ages of 12 students in a class. Here's your data: 15, 21, 19, 19, 20, 18, 17, 16, 17, 18, 19, 18.

Answer:

Highest value = 21

Lowest value = 15

Range = highest value - lowest value

Range = 21-15

Range = 6

However, the range has a few limitations:

It is affected by outliers.

It cannot be used for open-ended distribution.

A **quartile **is a type of quantile that divides an ordered data set into four parts (quarters). A quartile is not the group of numbers that have been divided. It is the cut-off point in the division.

**The interquartile range **is the difference between the upper quartile and the lower quartile value.

To find the quartile of a given data set you can proceed as follows:

Order the values in ascending order.

Find the median. This is always labeled as the second quartile ( ).

Now find the median of both halves of the data set. The lowest half is labelled , and the highest half is labelled .

Find the interquartile range (IQR) by subtracting Q1 from Q3.

Find the interquartile range for the data given 6, 9, 3, 6, 6, 5, 2, 3, 8.

Answer:

Reorder the values from lowest to highest.

2, 3, 3, 5, 6, 6, 6, 8, 9

Find the median

The median is 6.

= 6

Find the median of the two halves, which are: 2, 3, 3, 5 | 6, 6, 8, 9

For the first part, we have 3 as the median.

With the second step, we will have to sum both middle values and divide them by 2.

Find the interquartile range.

IQR = 7-3

IQR = 4

Variance and standard deviation are both measures of variability. The **variance **is the measure of how data points vary from the mean, and the **standard deviation **is the square root of variance. What this tells us is that standard deviation is derived from variance.

Variance is denoted by

Standard deviation is denoted by 𝝈.

The population variance formula is

Where = population variance

N = size of the population

= each value from the population

𝜇 = the population mean.

The sample variance formula is

Where = sample variance

n = size of sample

= each value from the sample

= the sample mean.

The population standard deviation formula is given by

Where 𝝈 = population standard deviation.

N = size of the population.

= each value from the population.

𝜇 = the population mean.

The sample standard deviation formula is given by

Where s = sample standard deviation.

n = size of sample.

= each value from the sample.

= the sample mean.

Calculate the standard deviation for the following scores on a Maths exam taken by -grade students: 82, 93, 98, 89, 88.

Answer:

The first thing you need to do is to find the mean of the sample:

So the formula that we are going to use here is , since the scores are available are only a sample of the whole population of students that took the exam.

We can construct a table to break down the formula and work it out appropriately.

82 | 64 | |

93 | 3 | 9 |

98 | 64 | |

89 | -1 | 1 |

88 | -2 |

According to the formula we will have to sum , which is the last column of our table.

s = 5,958

Standard deviation is 5.958

By definition, variance should be

- Statistical Measures are a technique of descriptive analysis used to give a summary of the characteristics of a data set.
- Measures of central tendency describe some key characteristics of the data set based on the average or middle values, as they describe the centre of the data.
- The three main measures of tendency are mean, mode, and median.
- Mean is the most common measure of central tendency and its formula is .
- Measures of spread are statistical measures that describe the similarity and variety of values of given datasets.
- Standard deviation is a measure of the amount of variation or dispersion of a set of values.
- Standard deviation is the square root of the variance.

Frequency distribution, standard deviation and mean.

A smaller standard deviation means greater consistency.

The extent to which two variables are linearly related.

More about Statistical Measures

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