Math Problem Statement

Consider a sample with data values of 10,20 ,12 ,17 , and 16 . Compute the z-score for each of the five observations (to 2 decimals). Enter negative values as negative numbers.

Solution

To compute the z-score for each observation, we use the following formula:

z=xμσz = \frac{x - \mu}{\sigma}

Where:

  • xx is an individual data value.
  • μ\mu is the mean of the data set.
  • σ\sigma is the standard deviation of the data set.

Step 1: Calculate the mean μ\mu

μ=10+20+12+17+165=755=15\mu = \frac{10 + 20 + 12 + 17 + 16}{5} = \frac{75}{5} = 15

Step 2: Calculate the standard deviation σ\sigma

First, find the squared differences from the mean for each data point:

(1015)2=(5)2=25(10 - 15)^2 = (-5)^2 = 25 (2015)2=(5)2=25(20 - 15)^2 = (5)^2 = 25 (1215)2=(3)2=9(12 - 15)^2 = (-3)^2 = 9 (1715)2=(2)2=4(17 - 15)^2 = (2)^2 = 4 (1615)2=(1)2=1(16 - 15)^2 = (1)^2 = 1

Now, sum these squared differences:

25+25+9+4+1=6425 + 25 + 9 + 4 + 1 = 64

Next, divide the sum by the number of data points minus 1 (this is the sample variance):

Variance=6451=644=16\text{Variance} = \frac{64}{5 - 1} = \frac{64}{4} = 16

Finally, take the square root of the variance to get the standard deviation:

σ=16=4\sigma = \sqrt{16} = 4

Step 3: Calculate the z-scores

Now, compute the z-score for each observation using the formula z=xμσz = \frac{x - \mu}{\sigma}.

For x=10x = 10:

z=10154=54=1.25z = \frac{10 - 15}{4} = \frac{-5}{4} = -1.25

For x=20x = 20:

z=20154=54=1.25z = \frac{20 - 15}{4} = \frac{5}{4} = 1.25

For x=12x = 12:

z=12154=34=0.75z = \frac{12 - 15}{4} = \frac{-3}{4} = -0.75

For x=17x = 17:

z=17154=24=0.5z = \frac{17 - 15}{4} = \frac{2}{4} = 0.5

For x=16x = 16:

z=16154=14=0.25z = \frac{16 - 15}{4} = \frac{1}{4} = 0.25

Final z-scores (rounded to 2 decimals):

  • z10=1.25z_{10} = -1.25
  • z20=1.25z_{20} = 1.25
  • z12=0.75z_{12} = -0.75
  • z17=0.50z_{17} = 0.50
  • z16=0.25z_{16} = 0.25

Would you like further clarification or additional details?

Here are 5 related questions:

  1. What does a z-score represent in statistics?
  2. How do z-scores relate to standard deviation?
  3. Why is it important to calculate z-scores in data analysis?
  4. Can you explain the difference between population and sample standard deviation?
  5. What does it mean if a z-score is negative?

Tip: Z-scores allow you to compare data points from different distributions by standardizing them.

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Math Problem Analysis

Mathematical Concepts

Statistics
Z-Score
Standard Deviation
Mean

Formulas

z = (x - μ) / σ
μ = (Σx) / n
σ = sqrt((Σ(x - μ)^2) / (n - 1))

Theorems

Z-score Theorem
Sample Standard Deviation

Suitable Grade Level

Grades 9-12