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Standardizing Data and Finding Z-Scores in Elementary Statistics, Summaries of Statistics

The concept of standardizing data and calculating Z-scores to compare values from different samples and identify unusual data elements. It includes formulas, examples, and solutions for finding Z-scores and determining ordinary and unusual values.

Typology: Summaries

2021/2022

Uploaded on 09/12/2022

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Elementary Statistics Z Score
Descriptive Statistics
Standardizing Data
Z–Score
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Descriptive Statistics

Standardizing Data

Z–Score

What is Standardizing Data?

Standardizing data is the process of putting different variables on the same scale. This process allows you to compare values from different samples such as exam results from different exams.

What does Standardization do?

Standardizing Data

◮ (^) produces the number of standard deviations above or below the mean that a specific observation falls, and ◮ (^) identifies the usual and unusual data element in the process.

What is x in the Z–Score formula?

It represents the data element that we want to Standardize.

What is x¯ in the Z–Score formula?

It represents the mean of the sample.

What is s in the Z–Score formula?

It represents the standard deviation of the sample.

Example:

Class exam had an average 78 with standard deviation of 6.8.

◮ (^) Find the Z score for exam result 90.

◮ (^) Find the data element associated with the Z score 2.5.

Solution:

◮ (^) For the Z score ⇒ we use the formula,

Z =

x − x¯ S

◮ (^) Z = x^ −^ x¯ S

x − 78

  1. 8

⇒ 2. 5 · 6 .8 = x − 78 ⇒ x = 95

Example:

John makes a monthly salary of $5750 as a nurse at the local hospital. The average salary for 25 randomly selected nurses was $5275 with standard deviation of $225. Find

◮ (^) Find the usual range of salaries according to the Z Score. ◮ (^) Find the Z–score for John’s salary. ◮ (^) Is John’s salary considered to be ordinary or unusual?

Solution:

The usual range ⇒ 5275 ± 2(225) ⇒ 4825 to 5725.

Z–score ⇒ Z =

x − ¯x S

Ordinary or unusual? ⇒ Unusual

Example:

Maria made 91 on exam 1 and 87 on exam 2 in her statistic class. Below is the summary of exam results for both exams.

Exam 1 Exam 2

¯x = 85 x¯ = 76. 8

s = 4. 8 s = 6. 8

◮ (^) Was any of her exam results unusual?

◮ (^) What exam did she do better?