Central Tendency and Probability:

What is Central Tendency?

Central tendency (sometimes called “measures of location,” “central location,” or just “center”) is a way to describe what’s typical for a set of data. Central tendency doesn’t tell you specifics about the individual pieces of data, but it does give you an overall picture of what is going on in the entire data set. There are three major ways to show central tendency: mean, mode and median.

Central Tendency Measures

Mean

The mean is the average of a set of numbers. Add up all the numbers in a set of data and then divide by the number of items in the set. For example, the mean of 2 3 5 9 11 is:

(2 + 3 + 5 + 9 + 11) / 5 = 30 / 5 = 6.

For more examples of finding the mean, see:

What is a mean?

Median

The median is the middle of a set of numbers. Think of it like the median in a road (that grassy area in the middle that separates traffic). Place your data in order, and the number in the exact center of a list is the median. For example:

1 2 3 4 5 6 7

The median is 4 because it’s in the center, with three numbers either side.

For more about the median, see:

What is a median?

Mode

The mode is the most common number in a set of data. For example, the mode of 1 2 2 3 5 6 is 2. Some data sets have no mode, like this one: 1 2 3 4 5 6. Others have multiple modes, like this one: 1 1 2 3 3.

For more on finding modes, see:

What is a Mode?

Outliers

Outliers are extremely high or extremely low values. Outliers can affect central tendency, especially the mean. For example, if you got paid three weeks in a row but took vacation in the fourth week, your paychecks might be: $300 $300 $300 $0. Your four week mean would be ($300 + $300 + $300 + $0) / 4 = $900/4 = $225. That outlier of zero dollars brought your mean down very low.

Source: http://www.statisticshowto.com/central-tendency-2/

 

Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

·        heads (H) or

·        tails (T)

We say that the probability of the coin landing H is ½

And the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is 16

Probability

In general:

Probability of an event happening =

Number of ways it can happen/Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Source: https://www.mathsisfun.com/data/probability.html

 

Multiple Events

Independent and Dependent Events

Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's.

We call two events independent if the outcome of one of the events doesn't affect the outcome of another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one- it's still 1/6.

On the other hand, suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of the bag, the probability that the second is blue depends upon what the colour of the first ball picked was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and 2 blue balls left so the probability the second ball is blue is 2/3. When the probability of one event depends on another, the events are dependent.

Source: https://revisionmaths.com/gcse-maths-revision/statistics-handling-data/probability

 

Helpful Links:

1.         Stats and Probability: Averages (has some quite advanced/complex analysis) https://www.wyzant.com/resources/lessons/math/statistics_and_probability/averages

2.         Simple Probability http://www.sparknotes.com/math/algebra1/probability/section1.rhtml

3.         Theoretical Probability http://www.basic-mathematics.com/theoretical-probability.html

Worksheet Links:

1.         Probability and Statistics Worksheets (lots of options) http://www.teach-nology.com/worksheets/math/stats/

2.         Determining Probability http://www.commoncoresheets.com/Probability.php

3.         Mean, Median, Mode & Range Worksheets http://www.commoncoresheets.com/MMMR.php

 

Video Links:

1.         Measures of Central Tendency https://www.youtube.com/watch?v=uRGi4Bb9RYQ

2.         Statistics Intro: Mean, Median, and Mode https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-center-distributions/v/statistics-intro-mean-median-and-mode

3.         Mean, Median, and Mode Example https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-center-distributions/v/mean-median-and-mode