Z-Score and Normal Distribution | Simplified Approach

 Z-Score and Normal Distribution

A Z-score is a statistical measure that tells you how far a particular data point is from the average (mean) of a group of data points, in terms of standard deviations. It's calculated by taking the difference between the data point and the average, and then dividing that by the standard deviation. A positive Z-score means the data point is above average (in the right direction), while a negative (in the left direction) one means it's below average. A Z-score of 0 means the data point is exactly at the average.

The normal curve, also known as the bell curve, is a symmetrical shape that many sets of data tend to form. It's called 'normal' because it's common in nature. In a normal distribution, the average, median, and most common values are all the same and sit right in the middle of the curve. The curve is symmetrical, forming a shape like a bell. The curve's width is determined by something called the standard deviation, which tells you how spread out the data is. Roughly 68% of the data falls within one standard deviation from the average, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations, that's by the empirical rule.

Z-scores are helpful because they let us compare data from different distributions. By converting raw scores into Z-scores, we can put everything on the same scale and make comparisons more straightforward. This is especially useful when dealing with different types of data or when trying to make sense of large sets of numbers.

What are the uses of Z Score?

Z-scores have various practical uses in statistics and data analysis.

1. Standardization. Z-scores allow for the standardization of data from different distributions, making it easier to compare values across different datasets. By converting raw scores into Z-scores, data points from different sources can be placed on a common scale, facilitating meaningful comparisons.

2. Outlier Detection. Z-scores are useful for identifying outliers in a dataset. Data points with Z-scores that fall significantly above or below the mean may be considered outliers and allow further investigation. This is particularly valuable in fields such as finance, where detecting anomalies in financial transactions or market behavior is crucial.

3. Probability Calculations. Z-scores can be used to calculate probabilities associated with specific values in a normal distribution. By referring to standard normal distribution tables or using statistical software, probabilities of observing values above or below a certain threshold can be determined.

4. Quality Control. Z-scores are employed in quality control processes to assess whether measured values fall within acceptable ranges. By setting thresholds based on Z-scores, deviations from expected values can be detected, signaling potential issues in manufacturing processes or product quality.

5. Performance Assessment. In fields such as education or sports, Z-scores are utilized to compare individuals' performances relative to their peers. By converting test scores or athletic performances into Z-scores, fair comparisons can be made, accounting for variations in the difficulty of assessments or competitions.

6. Risk Assessment. Z-scores play a crucial role in risk assessment models, particularly in the finance and insurance industries. They help quantify the level of risk associated with specific investments, loans, or insurance policies by measuring how far a particular value deviates from the mean in terms of standard deviations.

How to Easily Learn About Z-Scores

Understanding z-scores might seem tricky at first, but breaking it down step-by-step can make it much simpler. Here’s a straightforward way to get the hang of it:

Step-by-Step Guide to Z-Scores

1. Grasp the Basics of Statistics

   • Mean (μ). This is just the average of your data.
   • Standard Deviation (σ). Think of this as a measure of how spread out the numbers in your data set are.
   • Normal Distribution. Imagine a bell-shaped curve where most data points are clustered around the middle (the mean).

2. Learn the Z-Score Formula. Here’s the formula you’ll need: $z=\frac{X-\mu }{\sigma }$Where,

   • $X$ is the value you’re looking at.
   • $\mu$ is the average of your data.
   • $\sigma$ is the standard deviation.

3. What a Z-Score Actually Tells You

   • A z-score shows how many standard deviations a value is from the average.
   • Positive z-score- Above the average.
   • Negative z-score- below the average.
   • Z-score of 0- Exactly at the average.

4. Use Visual Aids

   • Graphs and Charts. Look at normal distribution curves to see where z-scores fit.
   • Z-Table. This table helps you understand the probability linked with each z-score.

5. Practice with Examples:

   • Start simple. Calculate z-scores with basic examples.
   • Move on to more complex, real-life scenarios to see how z-scores apply.

Practical Example

Your class took a test, and the scores are normally distributed with an average score of 80 and a standard deviation of 10. You scored 90. What’s your z-score?

Solution:

1. Identify your numbers:

   • $X=90$
     
   • $\mu =80$
     
   • $\sigma =10$
     

2. Plug them into the formula: $z=\frac{90-80}{10}=\frac{10}{10}=1$

3. Interpretation:

   • Your z-score is 1.
   • This means your score is 1 standard deviation above the average.

Tips for Learning

• Use Online Resources

  • Khan Academy. They have free courses and practice problems.
  • YouTube Tutorials. Sometimes a video can explain things better than text.

• Interactive Tools

  • Desmos Graphing Calculator. Helps you visualize normal distributions and z-scores.
  • Statistical Software. Tools like Excel, R, or Python (with libraries like NumPy and SciPy) are great for calculations and visualizations.

• Flashcards

  • Make flashcards with different z-score problems to quiz yourself.

• Study Groups

  • Discussing problems with friends can really help. Everyone might have a different way of understanding the concept.

• Real-World Applications

  • Think about z-scores in real life, like standardized test scores, quality control in products, or even in sports statistics.

Simple Z score problems with solutions (above or below the mean).

Problem 1

Suppose you have a class of 50 students who took a math test. The mean score on the test was 70, with a standard deviation of 10. If John scored 85 on the test, what is his Z-score?

Solution: To find John's Z-score, we'll use the formula:

$$

​

Where,

• $x$ is John's score (85)
• $\mu$ is the mean score (70)
• $\sigma$ is the standard deviation (10)

Substituting the values into the formula:

$$

Therefore, John's Z-score is 1.5.

Interpretation

A Z-score of 1.5 means that John's score is 1.5 standard deviations above the mean score of the class. This indicates that John performed well above the average compared to his classmates on the math test.

Normal Curve visualization (above the mean).

Problem 2

Suppose you have a group of students who took a standardized test, and the test scores follow a normal distribution with a mean score of 75 and a standard deviation of -10. If Jane scored 85 on the test, what is her Z-score?

Solution: To find Jane's Z-score, we'll use the formula:

​​

Where:

• $x$ is Jane's score (85)
• $\mu$ is the mean score (75)
• $\sigma$ is the standard deviation (-10)

Substituting the values into the formula:

$$

So, Jane's Z-score is -1.

Interpretation

A Z-score of -1 means that Jane's score is 1 standard deviation below the mean score of the group. This indicates that Jane performed slightly below the average compared to her peers on the standardized test.

Normal Curve visualization with its area using a z-table (below the mean).

Problem 3

A group of 500 students took a standardized math test. The test scores are normally distributed with a mean ($\mu$) of 75 and a standard deviation ($\sigma$) of 10. John scored 85 on the test.

1. What is John's z-score?
2. What percentage of students scored below John?

Solution:

Calculating John's z-score:

The z-score is calculated using the formula:

$$$$

where $X$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Plugging in the values

$$$$

So, John's z-score is $1$.

Interpretation

A z-score of 1 means that John's score is 1 standard deviation above the mean. In other words, John scored better than the average student by an amount equal to one standard deviation. Since the mean test score is 75 and the standard deviation is 10, John's score of 85 is exactly 10 points higher than the average score.

Normal Curve visualization with its area using a z-table (below the mean).

Solved problems

Problem 1

A local college conducted a survey to determine the average number of hours students spend studying per week. The survey sampled 50 students and found an average study time of 16 hours per week with a standard deviation of 3 hours. If a student from this college studies 20 hours a week, what is their z-score?

Solution

To find the z-score, we use the formula:

$z=\frac{X-\mu }{\sigma }$

Where:

• $X$ is the value of the data point (in this case, the number of hours the student studied, which is 20).
• $\mu$ is the mean (the average number of hours students study per week, which is 16).
• $\sigma$ is the standard deviation (3 hours in this case).

Plugging in the values, we get.

$z=\frac{20-16}{3}$

$z=\frac{4}{3}$

$z\approx 1.33$

Interpretation

A z-score of 1.33 means that the student who studied 20 hours per week is 1.33 standard deviations above the mean study time for the surveyed students. This indicates that the student studies more than the average student at the college, and the difference is slightly more than one standard deviation above the average.

Problem 2

A high school conducted a math test and found that the scores were normally distributed. The mean score is 75 with a standard deviation of 10. One student scored 85 on the test. What is the z-score for this student's score?

Solution

To find the z-score, we use the formula:

$z=\frac{X-\mu }{\sigma }$

Where:

• $X$ is the student's score (85).
• $\mu$ is the mean score (75).
• $\sigma$ is the standard deviation (10).

Plugging in the values

$z=\frac{85-75}{10}$

$z=\frac{10}{10}$

$z=$

Interpretation 

The z-score of 1 means that the student's score is 1 standard deviation above the mean. This student's performance is better than the average student's performance in this test.

Problem 3

In a university, the average GPA of graduating students is 3.2 with a standard deviation of 0.4. If a particular student's GPA is 3.8, what is their z-score?

Solution 

To calculate the z-score:

$z=\frac{X-\mu }{\sigma }$

Where:

• $X$ is the student's GPA (3.8).
• $\mu$ is the average GPA (3.2).
• $\sigma$ is the standard deviation (0.4).

Plugging in the values:

$z=\frac{3.8-3.2}{0.4}$

$z=\frac{0.6}{0.4}$

$z=1.5$

Interpretation

A z-score of 1.5 indicates that the student's GPA is 1.5 standard deviations above the average GPA. This student is performing significantly better than the average student.

Problem 4

A fitness center analyzed the weekly exercise time of its members. The average weekly exercise time is 150 minutes with a standard deviation of 30 minutes. One member exercises for 90 minutes per week. What is the z-score for this member's exercise time?

Solution

To determine the z-score:

$z=\frac{X-\mu }{\sigma }$

Where:

• $X$ is the member's exercise time (90 minutes).
• $\mu$ is the average exercise time (150 minutes).
• $\sigma$ is the standard deviation (30 minutes).

Plugging in the values:

$z=\frac{90-150}{30}$

$z=\frac{-60}{30}$

$z=-2$

Interpretation

A z-score of -2 means that the member's exercise time is 2 standard deviations below the mean. This member exercises significantly less than the average member at the fitness center.

Problem 5

A company's annual employee performance scores are normally distributed with a mean of 70 and a standard deviation of 8. If an employee received a performance score of 62, what is their z-score?

Solution: To find the z-score, we use the formula:

$z=\frac{X-\mu }{\sigma }$

Where:

• $X$ is the employee's performance score (62).
• $\mu$ is the mean performance score (70).
• $\sigma$ is the standard deviation (8).

Plugging in the values:

$z=\frac{62-70}{8}$

$z=\frac{-8}{8}$

$z=-1$

Interpretation

The z-score of -1 indicates that the employee's performance score is 1 standard deviation below the mean. This suggests that the employee's performance is below the average performance level of employees at the company.

Practice test

1.  A class of 30 students took a history exam. The scores are normally distributed with a mean of 70 and a standard deviation of 8. Emily scored 78 on the exam.

Ø  What is Emily's z-score?

Ø  What percentage of students scored below Emily?

2. The weights of apples in an orchard are normally distributed with a mean of 150 grams and a standard deviation of 20 grams. An apple weighs 180 grams.

Ø  What is the z-score for this apple's weight?

Ø  What percentage of apples weigh less than 180 grams?

3. A company's employee satisfaction scores are normally distributed with a mean of 60 and a standard deviation of 15. One employee scored 45 on the satisfaction survey.

Ø  What is the z-score for this employee's satisfaction score?

Ø  What percentage of employees scored higher than this employee?

4. The heights of adult males in a certain region are normally distributed with a mean of 175 cm and a standard deviation of 10 cm. A man is 190 cm tall.

Ø  What is the z-score for this man's height?

Ø  What percentage of men are shorter than 190 cm?

5. The reaction times of participants in a psychological experiment are normally distributed with a mean of 300 milliseconds and a standard deviation of 50 milliseconds. A participant has a reaction time of 250 milliseconds.

Ø  What is the z-score for this participant's reaction time?

Ø  What percentage of participants have a slower reaction time than 250 milliseconds?

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