Z-Scores Table

A Comprehensive Overview of the Z-Scores Table

1. An Overview of Statistics Standardization

Data used in statistics frequently originates from a variety of sources and takes several forms, including test results, weights, heights, production measurements, economic indicators, and more. These values could be based on several scales or units. For instance, a person might receive a score of 85 out of 100 on one standardized examination and 650 out of 800 on another. How can we make meaningful comparisons between these values?

Statisticians employ a procedure known as standardization to address this. The Z-score is the most widely used technique for data standardization.

2. Z-Score Table: What Is It?

The relationship between a data point and the group mean is expressed numerically by a Z-score. More specifically, it indicates the number of standard deviations that separate a data point from the mean.

The value is exactly average—it falls at the mean—if the Z-score is 0.

The value is above the mean when the Z-score is positive.

The value is below the mean when the Z-score is negative.

Z-scores table enable us to comprehend a value's position within its data set as well as the extent to which it is exceptional or typical.

 

Z-Scores Table

First column = Z-score to one decimal (row), top row = hundredths (column)

Z-Scores	.00	.01	.02	.03	.04	.05	.06	.07	.08	0.09
-3.4	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003
-3.3	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005
-3.2	0.0007	0.0007	0.0007	0.0007	0.0007	0.0007	0.0007	0.0007	0.0007	0.0008
-3.1	0.001	0.001	0.001	0.0011	0.0011	0.0011	0.0012	0.0012	0.0013	0.0013
-3.0	0.0013	0.0013	0.0013	0.0014	0.0014	0.0015	0.0015	0.0016	0.0016	0.0017

 

Get the full Z-score table here

 

1. The Z-Score Formula

The following formula can be used to determine a data value's Z-score:

Where:

• Z = Z-score (standardized score)

• X = the original raw value

• μ (mu) = the mean (average) of the distribution

• σ (sigma) = the standard deviation of the distribution

Any value from any normal distribution can be converted into a value on the standard normal distribution using this formula, which has the following properties:

• A mean of zero

• One standard deviation

2. What Makes Z-Scores Table Vital?

Z-scores table is employed in numerous data analysis domains, such as: 

a) Scale-to-Scale Comparison

Assume that two pupils take distinct exams:

On a math test, Alice receives an 85 (mean = 75, SD = 5).

Bob's SAT score is 650 (mean = 600, SD = 30)

The raw scores cannot be directly compared. However, their Z-scores can be compared: 

• Alice:

• Bob:

In conclusion, Alice outperformed her group because her score was two standard deviations higher than Bob's, which was 1.67 standard deviations.

b) Recognizing Outliers Unusual or outlier values are those that have extremely high or extremely low Z-scores (usually |Z| > 2 or |Z| > 3).

c) Percentiles and Probabilities

The basis for determining probabilities in a normal distribution is Z-scores table. For instance, a Z-score table (examined in later sections) can be used to ascertain the probability that a particular score or event will occur.

d) Making Distributions Standard

Numerous distributions with varying means and variances can be compared on a single standardized scale thanks to Z-scores.

 

Understanding Mean and Standard Deviation

To completely grasp Z-scores table, it's vital to understand the components in the formula:

a) Average (μ)

The mean is the dataset's average value:

where nn is the number of data points and ∑X is the sum of all data values.

b) σ (standard deviation)

The degree of dispersion of the data around the mean is indicated by the standard deviation. When the standard deviation is low, the data is concentrated near the mean; when it is large, the data is widely dispersed.

To ascertain a value's position within its distribution, one must consider both the mean and the standard deviation.

 

Interpreting Z-Scores

Let’s break down what a Z-score tells us:

• Z = 0
  The value is exactly at the mean.

• Z = +1
  The value is 1 standard deviation above the mean.

• Z = +2
  The value is 2 standard deviations above the mean (somewhat unusual).

• Z = -1
  The value is 1 standard deviation below the mean.

• Z = -3
  The value is 3 standard deviations below the mean (rare or exceptional).

General rule of thumb:

• |Z| < 1 → very typical

• 1 ≤ |Z| < 2 → somewhat uncommon

• |Z| ≥ 2 → unusual

• |Z| ≥ 3 → very rare/extreme

 

Real-World Instances: Z-score formula

First Example:

A student receives a test score of 90. The standard deviation is five, while the test mean is eighty.

The student performed well, scoring two standard deviations above average.

Example 2: Temperature of the Body

Assume a standard deviation of 0.7°F and an average body temperature of 98.6°F. A temperature of 100.1°F would result in:

Given that this temperature is little over two standard deviations higher than normal, a moderate fever may be suspected.

 

Conclusion

A fundamental idea in statistics and data analysis is Z-scores. They offer a straightforward yet effective method of comprehending the relationship between a single data point and the remainder of the distribution. Z-scores enable meaningful comparisons across various data sets, units, and contexts by transforming raw values into standardized scores. Exam scores, medical test results, and manufacturing quality can all be evaluated using Z-scores, which show how common, uncommon, or important a certain observation is. Gaining proficiency with the Z-score formula and its interpretation paves the way for a deeper comprehension of inferential statistics, probability, and distribution analysis.

 

Questions for Self-Check

1. What is meant by a Z-score of -1.5?

2. What does it tell you if a student's Z-score is zero?

3. If the population mean is 170 cm and the standard deviation is 5 cm, find the Z-score for a height of 180 cm.

4. Is a negative Z-score possible? What does it mean?

5. Why is it preferable to compare test results using Z-scores rather than raw scores?

 

Related links:

• Inches to CM Converter – Formula And Examples

• Compound Interest Formula: Calculator, Chart & Table Guide

 

FAQs

 

1. What is 95%'s Z-score?

In a conventional normal distribution, a Z-score of 95% depends on whether you're requesting:

Left-tailed cumulative probability: About +1.645 is the Z-score that places 95% of the region to the left.

Middle 95% (central coverage): The Z-scores are roughly ±1.96 in order to cover 95% in the middle of the distribution, which leaves 2.5% in each tail.

Confidence intervals and hypothesis testing frequently employ these values.

 

2. What is the purpose of Z-score tables?

The area (or cumulative probability) to the left of a given Z-score in the standard normal distribution is displayed in a Z-score table, also known as the standard normal table. It enables you to:

Determine the likelihood that a score will fall below (or rise over) a specific value.

Z-scores and percentiles can be converted.

Resolve issues with normal distributions and probability.

For instance, 84.13% of values fall below a Z-score of 1.00, which is equivalent to a cumulative probability of 0.8413.

 

3. What does a Z-score of two mean?

The data value is two standard deviations above the mean when the Z-score is 2. It shows that, under a normal distribution, the value is higher than the average:

A Z-score of 2 is below about 97.72% of values.

Approximately 2.28% of values are higher than it.

Although it would be seen as odd, this Z-score is not very uncommon.

 

4. What is a Z-score that is healthy?

In general, the context determines what constitutes a "healthy" Z-score.

In general statistics, a Z-score that falls between -2 and +2 is regarded as being within the average or normal range. In a normal distribution, this range encompasses the majority of data (about 95%).

In health or medical contexts (bone density, for example):

For BMD, or bone mineral density:

Z-scores above -2.0 are often regarded as normal.

In younger people in particular, a Z-score below -2.0 may suggest a possible underlying health problem.

As a result, a healthy Z-score is one that is neither abnormally high nor low, but rather represents being near the population average.