Statistics Part #1Measures of Central Tendency

Measures of Central Tendency

a measure of the balance of an ordered data set

the most common measures of central tendency are mean, median, and mode.

Arithmetic Mean or Mean

the sum of a set of values divided by the number of values; you may know this as the “average”

Real Life Applications:

a teacher uses the mean to calculate the class average on a test

a weather reporter uses the mean to calculate the average temperature in a month

ExampleCalculate the mean of the following set of data:

5, 6, 8, 3, 4, 6, 9, 2, 4, 5, 6, 7, 9

Hint: Cross off numbers as you do the calculations and double check to ensure

you have all numbers included

ExampleA grade 12 class with 15 students was polled to determine

the number of television sets they had in their homes. These are the results:

3, 4, 1, 3, 2, 2, 1, 1, 4, 5, 8, 3, 2, 7, 4

What is the mean of this set of data?

Challenge YourselfMs. James compiled a list of her classes test scores.

The mean test score is 23 out of 50. Calculate the unknown score:

9 15 3527 21 1631 19 4212 26 4832 20 15 13 x

Challenge YourselfMs. James compiled a list of her classes test scores.

The mean test score is 23 out of 50. Calculate the unknown score:

Step 1: Add up all of the numbers that you can

Step 2: Use your knowledge of variables to solve for “x”

Step 3: Check your work by plugging your value into the data set to ensure the statement is true.

Challenge YourselfSolve for the missing data point.

The mean is 36.

72 30 52 40 2512 25 16 51 x

Challenge YourselfSolve for the missing data point.

The mean is 4.586

3.14, 2.71, 10.27, 5.24, x

Median

the midpoint of an ordered data set; half the values are higher and half are lower

if there is an odd number of values in the list, the middle value is the median; if there is an even number of values, then the arithmetic mean of the two middle values is the median

Median

Real Life Applications:

the median is referred to when sales of houses are reported; the median selling price in a city tells you that half the houses sold at a higher price and half at a lower price

the median is often used to report typical wages paid to employees

ExampleFind the median of this set of numbers:

5, 6, 8, 3, 4, 6, 9, 2, 4, 5, 6, 7, 9

First, you must organize them from least to greatest:

ExampleA grade 12 class with 10 students was polled to determine

the number of television sets they had in their homes. These are the results:

3, 2, 7, 6, 2, 1, 3, 2, 4, 1

What is the median of this set of data?

Mode

in a list of values, the value that occurs the most often is called the mode.

If each value occurs only once, there is no mode

If two or more values appear the same number of times and it is the greatest number of occurrences in the data set, then they are all considered the modes.

two modes = bimodal; three modes = trimodal; etc.

ModeReal Life Applications:

if you are a shoe manufacturer and you know more women wear size 7 shoes than any other size, you may use the mode to set your production run

if you manage a building supply store and your customers buy more 2” by 4” pieces of dimensional lumber than any other size, you will order your stock accordingly

Example

Find the mode in the following set of data

5, 6, 8, 3, 4, 6, 6, 9, 2, 4, 5, 7, 9

First, we must order these values from least to greatest

ExampleA grocery store kept record of how many plastic shopping bags their customers used per visit, here are the results.

1, 5, 3, 4, 2, 4, 3, 1, 3, 2, 4, 6

Determine the mode:

Try These!1. What is the mean, median and mode for this number set?

[13, 14, 12.5, 14, 15, 16]

2. A class with 15 students was polled to determine the number of family members they had living in their homes.

These are the results:

2, 3, 7, 4, 2, 8, 4, 4, 3, 2, 5, 6, 9, 12, 14

What is the mean, median and mode of this set of data?

Try These!

3. The mean for the following set of data is 42.5, calculate the missing score.

14, 20, 15, 19, 70, 10, 36, 26, 45, 29, 41, 63, 18, 31, 55, 61,

73, 82, 90, x

then, calculate the median and the mode