Name______Block______Date______

Statistics Lesson 1 Mean Review, Standard Deviation & Z-scores

In a park that has several basketball courts, a student counts the number of players playing basketball each day over a two week period and records the following data.

10, 90, 30, 20, 50, 30, 60, 40, 70, 40, 30, 60, 80, 20

In another park that has several basketball courts, another student counts the number of players playing basketball each day over a two week period and records the following data.

50, 40, 30, 30, 40, 50, 50, 30, 40, 50, 60, 60, 50, 50

How are the two data sets similar and how are they different?

Mean () data set #1 = Mean () data set #2 =

Median data set #1 = Median data set #2 =

Mode data set #1 = Mode data set #2 =

Range data set #1 = Range data set #2 =

Practice

1.  Find the mean for this data.

12, 15, 19, 17, 18, 17, 20, 14 mean______

2.  If the mean of a set of 12 number is 13, then the sum of the numbers is: ______

Standard Deviation

Statisticians like to measure the dispersion (spread) of the data set about the mean in order to help make inferences about the population.

The greater the value of the standard deviation, the more spread out the data are about the mean. The lesser (closer to 0) the value, the closer the data are clustered about the mean ( ).

- an element of a data set where represents the th term of the data set.

- the mean of a data set

n - the number of elements in the data set

Standard Deviation () =

Using the TI-84+ to find Standard Deviation:

Enter the data into L1

• From the home screen click on STAT.

• To enter data into lists choose option “1:Edit”

• In L1, enter each element of the data set.

Enter the first element and press enter to move to the next line and continue for all elements.

Calculate standard deviation by computing 1-Variable Statistics for L1

• Press STAT

• Press the Right Arrow (highlight CALC)

• Choose option “1: 1-Var Stats”

• Press enter to compute the 1-variable statistics (defaults to L1).

= arithmetic mean of the data set

Σx = sum of the x values

Σx2= sum of the x2 values

Sx = sample standard deviation

σ x = population standard deviation

n = number of data points (elements)

The following are heights (in inches) from the class: 66, 62, 68, 63, 72, 69, 59

Calculate the Standard Deviation.

- 1 deviation = ______=______+ 1 deviation = ______

50 55 60 65 70 75

Which x values lie within one standard deviation of the mean?______

A z-score is the number of standard deviations that a given x-value lies above or below the mean.

z-score (z) =

A student with the height of 72 inches has a z-score of

The student’s height was 1.54 standards away from the mean. The z-score has a positive value if the element lies above the mean and a negative value if the element lies below the mean.

Practice:

1.  On six consecutive Sundays, a tow-truck operator received 9, 7,11,10,13, and 7 service calls. Calculate the standard deviation.

2.  Heights of adult males have a mean of 69 and a standard deviation of 2.8 inches. Basketball player Michael Jordan earned a giant reputation for his skills, but at a height of 78 inches, is he exceptionally tall when compared to the general population of adult males? Find the z-score for his height. Interpret the result.

Stats Lesson 1 Homework Name______

The number of shots made from 50 free throws for 18 players are counted.

1. Find the: Mean…Median…Mode…Range… for this set of data

Mean: ______Mode: ______

Median: ______Range: ______

2. Find the Standard Deviation of this data set.

3. If Jake made 40 of the 50 free throws, how many standard deviations away from the mean is this (what is his z-score)?

Comparing Standard Deviations

Let's look at this concept graphically. If I have a large standard deviation, it indicates that my scores are widely dispersed, and therefore, my Mean doesn't really tell me that much about the average score. On the other hand, if my standard deviation is small, it indicates that my scores are close to the Mean, and therefore, the Mean is a good indicator of the "average" score.

4.

Use the information below to answer questions 5-8

5. How many elements are below the mean? ______
6.  How many elements are above the mean? ______
7.  What is the range of heights from -1 to 1 standard deviations? ______
8.  How many elements fall within one standard deviation of the mean? ______
9.  Find the z-score corresponding to a score of 89 from a normal distribution with mean 81 and standard deviation 7.
z-score =
10.  How many standard deviations away from the mean is 56, if the mean is 70 and the standard deviation is 6?
z-score =
11.  The average dog is 394 cm tall. The Robo the Rottweiler is 630cm tall. How many standard deviations away from the mean is Robo? The standard deviation is 40 cm.
z-score =
12.  What does a negative z-score tell you?