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Measures of Central Tendency, Dispersion, IQR and Standard Deviation. How do we describe data using statistical measures?. M2 Unit 4: Day 1. Statistics: numerical values used to summarize and compare sets of data.

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measures of central tendency dispersion iqr and standard deviation

Measures of Central Tendency, Dispersion, IQR and Standard Deviation

How do we describe data using statistical measures?

M2 Unit 4: Day 1

statistics numerical values used to summarize and compare sets of data
Statistics: numerical values used to summarize and compare sets of data.

Measure of central tendency: a number used to represent the center or middle of a set of data values.

We will use 3 types:

  • Mean
  • Median
  • Mode
slide3
Also called the average

given n numbers, it is the sum of the n numbers divided by n

is the sample mean

is the population mean

Mean:
example
Example
  • Find the mean of the set of data.

75, 100, 63, 95, 82, 78

median
Median:

given n numbers, it’s the middle number when written in ascending order (from least to greatest)

n can be odd or even

example1
Example
  • Find the median of the data set.
  • 1. Case 1(n is odd)

100, 95, 50, 78, 63, 75, 82

50, 63, 75, 78, 82, 95, 100

78 is the median

example2
Example
  • Find the median of the data set.
  • 2. Case 2 (n is even)

82, 63, 100, 75, 78, 95

63, 75, 78, 82, 95, 100

you try find the median
You try: Find the median

3. 12, 13, 12, 10, 11, 14, 10

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

slide9
Mode

the number that occurs most frequently in a given data set

There are 3 cases:

1 Mode

No Mode

More than one Mode

example find the mode
Example: Find the mode

1. Find the mode of the data set: 1, 2, 3, 4, 5, 3

3 is the mode

2. Find the mode of the data set: 1, 2, 3, 4, 5

There is no mode

3. Find the mode of the data set: 1, 2, 3, 4, 5, 2, 3

2 and 3 are the modes

measure of dispersion
Measure of dispersion:

is a statistic that tells you how dispersed (spread out) the data values are.

One example of a measure of dispersion is range.

Range is difference between the largest and smallest data values

example3
Example:
  • Find the range of the data set:
  • 63, 75, 78, 82, 95, 100

100 - 63 = 37

  • Find the range of the data set:
  • 21, 20, 26, 30, 16, 20

30 – 16 = 14

interquartile range iqr
Interquartile Range (IQR)

The distance between the first and third quartiles

To calculate, find the median of the upper and lower half, then take the difference

example4
Example

1. Find the IQR: 100, 78, 63, 50, 82, 95, 75

50, 63, 75, 78, 82, 95, 100

Find the median

Find the 1st and 3rd quartiles

IQR = 95 - 63

= 32

you try
You Try:

Find the IQR: 78, 83, 91, 81, 111, 83, 72

72, 78, 81, 83, 83, 91, 111

Find the median

Find the 1st and 3rd quartiles

IQR = 91 - 78

= 13

example5
Example

Find the IQR: 100, 63, 75, 82, 95, 100, 50, 78

50, 63, 75, 78, 82, 95, 100, 100

you try1
You Try:

Find the IQR: 2, 3, 2, 4, 1, 8, 5, 6

1, 2, 2, 3, 4, 5, 6, 8

population standard deviation sigma
Population Standard Deviation( “sigma”):
  • measures the spread by looking at how far the observations are from their mean.
  • The smaller the standard deviation, the less the data varies about the mean .
  • The larger the standard deviation, the more the data will vary about the mean.
  • is the population standard deviation
  • is the sample standard deviation
example6
Example:

Find the standard deviation for the following data set: 2, 5, 7, 11, 15

Find the standard deviation of the sample: 3, 4, 8, 9, 10

example7
Example

A sample of 6 temperatures of patients was taken from all of the patients on wing E of the hospital.

The temperatures are:

98.6, 101, 97.8, 98, 99.4, 100.1

What is the standard deviation?

example8
Example

A teacher looked at the GPAs of her advisement group.

The GPAs are:

95.3, 91.2, 86, 90.2, 82.2, 70.1, 72.3, 68.1, 75

Is the a sample or a population?

What is the standard deviation?

example9
Example

A sample of 8 prices was taken from the menu of a given restaurant.

The prices are:

$3.25, $10.75, $0.75, $2.00, $1.50, $8.45, $6.00, $4.45

Is the a sample or a population?

What is the standard deviation?

example10
Example

The following prices are for entrance into different sporting events at a given school.

The prices are:

$4.00, $2.50, $3.00, $5.00, 7.50

Is the a sample or a population?

What is the standard deviation?

example compare the mean and standard deviation for the number of cars sold by the 2 dealers
Example :Compare the mean and standard deviation for the number of cars sold by the 2 dealers
  • Dealer A: 8, 9, 15, 25, 20, 16, 24, 18, 21, 14, 16, 10
    • mean = 16.3 ; standard deviation = 5.34

Dealer B: 7, 4, 10, 18, 21, 30, 27, 20, 16, 18, 12, 9

mean = 16; standard deviation = 7.6

On average, Dealer A sells more cars per month than Dealer B. Dealer A has a smaller standard deviation than Dealer B. Therefore, the amount of cars hat Dealer A sell from month to month varies less than that of Dealer B. That is, Dealer A is more consistent in the number of cars he sells than Dealer B.

practice problem 1
Practice Problem #1

Directions: Find the Mean, Median, Mode, Range, IQR and Standard Deviation for the following data set.

  • mean: 52.42
  • median: 53
  • mode: No Mode
  • Range: 39
  • IQR: 24
  • SD: 12.62
slide26
If the average monthly temperature increased by 2 degrees each month, how would this affect the mean and standard deviation?
  • The mean would increase by 2 and the standard deviation would remain the same.
slide27
If the average monthly temperature increased by 3 degrees in the Month of March and July, how would this change the mean?

The mean and the standard deviation would increase.

practice problem 2
Practice Problem #2
  • Directions:
    • Find the mean value of the shutouts.
    • Find the interquartile range.

mean: 72.8

IQR: 19

practice problem 3
Practice Problem #3

Directions: The following is a list of lengths (in minutes) of 13 Movies. Find the Mean and Standard Deviation for the following data set.

90, 102, 120, 180, 90, 85, 90, 137, 120, 151, 97, 93, 120

mean: 113.46

SD: 27.46

practice problems 4
Practice Problems # 4

The following is a list of lengths (in minutes) of 13 Movies:

90, 102, 120, 180, 90, 85, 90, 137, 120, 151, 97, 93, 120

If all movie times increased by 10 minutes, how would the mean and standard deviation be affected?

Mean would increase by 10 and the standard deviation would remain the same.

slide31
Assignment:
    • Day 1 Handout
    • Review packet