Mathematics Domain 2 Competency 17

Probability and Statistics

What’s the goal of this competency?

• The teacher understands concepts related to probability and statistics and their applications.

Key Definitions

Statistics – The branch of mathematics that deals with the collection, organization, analysis, and interpretation of numerical data. It is the science or the study of data.

Probability- The ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes. It gives us a way to measure uncertainty.

How do probability and statistics relate to each other?

DATA ANALYSIS

– Involves both probability and statistics.

Examples of probability in the real world.

- Weather Forecasting

- Batting Averages

- Winning the Lottery

- Insurance premium calculation

Data Set Descriptors 1

• Range -The range of a set of data is the difference between the highest and lowest values in the set. To find the range, first order the data from least to greatest. Then subtract the smallest value from the largest value in the set.

Example

Problem: Cheryl took 7 math tests in one marking period. What is the range of her test scores?  89,  73,  84,  91,  87,  77,  94

Solution:  Ordering the test scores from least to greatest, we get:  73,  77,  84,  87,  89,  91,  94  highest - lowest = 94 - 73 = 21

Answer:  The range of these test scores is 21 points.

Data Set Descriptors 2

• Mean – The mean of a set of data is found by taking the sum of the data and dividing by the total number of values in the set. The mean is commonly referred to as the average.

Example

Problem:  Scott took 7 math tests in one marking period. What is the mean test score?  89,  73,  84,  91,  87,  77,  94

Solution:  The sum of these numbers is 595. Dividing the sum by the number of test scores we get:   595 divided by 7

Answer:  The mean test score is 85.

Data Set Descriptors 3

• Median-The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest.

Example

Problem: The Doran family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?

Solution: Ordering the childrens' ages from least to greatest, we get:  7,  9,  12,  13,  16 Answer:  The age of the middle child is the middlemost number in the data set, which is 12.

Data Set Descriptors 4

• Mode- The mode of a set of data is the value in the set that occurs most often.

Example

Problem:  The number of points scored in a series of football games is listed below. Which score occurred most often? 7,  13,  18,  24,  9,  3,  18

Solution:  Ordering the scores from least to greatest, we get:  3,  7,  9,  13,  18,  18,  24

Answer:  The score which occurs most often is 18.

Basics for Elementary School Students

• Concert examples are better for this age group, the students need real problems and/or simulations.

Steps for an Experiment

1. Data Collections

2. Sampling

3. 3.Organizing and Representing Data

4. Interpreting Data

5. Assigning Probabilities

6. Making Inferences

Probability Formula

• Probability is a way of describing how likely it is

a particular outcome will occur.

• Probability results (fraction)=

Number of favorable outcomes (numerator)

Total number of possible outcomes (denominator)

Coin Toss Example

Probability –Gauge of Understanding

1) Use of experimental and theoretical probability to make predictions.

2) Use of statistical representations to analyze data.

Sample Set

The set of all possible outcomes of an experiment.

Permutations

All possible arrangements of a given number of items in which the order of the items make a difference.

Ex: Different ways a set of four books can be placed on a shelf.

Jones, Langral, Thornton and Mogill

4 Stages of the Learning Process –Probability

Subjective Level- Learners easily swayed by personal experiences when making probabilistic statements.

Second Level- Transitional learners begin to organize the importance of organizing information.

Third Level- Students begin to become informal quantitative thinkers.

Numerical Level- Students understand the nuances of numerical argument and use sophisticated procedures to determine numerical facts.

Sample Lesson

Data Displays

Students will utilize and interpret:

• Tables

• Bar Graphs

• Circle Graphs

• Line plot

• Pictographs

to compare data through finding the means, medians, and modes of the information presented.