Z Table

There is a varied range of values while analyzing huge data sets. For instance, people’s weight can range from a few pounds to five hundred pounds, people’s height varies from three feet up to eight feet.

Therefore, standardizing the data and setting a reference point becomes essential. A normal distribution curve is used to represent the entire data set and the distribution of data at different points. Furthermore, the z-table is used to determine the percentage of data under the curve at a specific point.

In this article, we will understand what a z-table is and what is meant by the positive or negative z-score table. We will practice a few questions too. Finally, we will glance at the history of standard normal deviation.

Part 1: What is Z Table

A ‘z-table’ is a short form for ‘Standard Normal z- table‘. It is a commonly used table in statistics and data sciences. Z-table is an important tool for testing of hypothesis and analysis of data.

A z-table is used to map values against the points distributed throughout the standard normal curve. The area below the curve represents the whole set of data and how data is distributed at various points.

The z-table tells us about the standard deviation from the mean value, as we move towards or away from the reference point. A z-table gives you clarity about the performance and score of anything at a single glance. You can easily know how good or bad a metric is, as compared to the other values in the same data set.

The central point of the curve is the reference point ‘Mean Value’ and labeled as ‘0’.The curve is divided into 6 parts, referred to as z-score -3, -2, -1, 0, 1, 2, 3. The values on the right side are positive, whereas negative values reside as we move to the left of the curve.

A z-table gives precise values of standard deviation at a given point. Z-table at point 0 is 50% or “0.5” z-score. It follows a rule of 68, 95 and 99.7. Approximately 68% of data falls below the range of -1 to +1, 95% falls below the range of -2 to +2 and 99.7% below the range of -3 to +3.

While looking at the mapping of z-score on to the standard distribution curve, you can find three distinct areas:

  • The area under the curve: You can observe different properties of normal distribution from the area under the curve. You can see how many values fall under the predetermined limits.
  • The area Between The Mean And The Z: This area is the area below the curve, between the mean and the z-scores. You can see the ratio of scores between the mean and any given z-score in this area.
  • The Area Beyond Z: You will see the proportion of scores higher than any given z-score in this area.

Part 2: Negative Z score table

The left-hand tail of the normal standard deviation curve contains the negative values. The negative z-score table is used to find values on the left side of the mean. These values are basically less than the mean. Hence, they are signed as ‘negative’.

Simply put, a negative z-score tells us that the particular value is below the mean value.

See the full negative z-score table below:

negative ztable

Part 3: Positive Z score table

Positive z-score indicates the values which are higher than the mean. These values reside on the right-hand side of the curve. The z-score is positive when you move away from the central point of the mean value.

A positive z-score tells the value of the metric is higher than the mean.

See the full positive z-score table below:

postive ztable

Part 4: Origin and Development of Standard Normal Distribution

The origin of Standard Normal Distribution can be tracked down to the 16th century. It is interesting how French mathematician Abraham de Moivre applied the principles of maths to crack success in gambling. He provided consultation in the gambling and became inquisitive about the likelihood of getting a top in a certain number of coin flips.

In an effort to define a mathematical formula for calculating the probability of success, he discovered the concept of distribution curve. The data sets contain a wide range of values which can be standardized against a mean value of zero. This mean value corresponds to a standard deviation of 1. The normal distribution curve told him about the chances of getting the desired result.

Later on, mathematicians realize that this normal distribution is applicable to various real life and mathematical phenomena. Lambert Quetelet, a Belgian astronomer, was the first to notice the connection between the distribution of weight and height and the normal curve.

In astronomical observations and calculations, Galileo discovered that the errors were symmetric in nature.  It was realized in the nineteenth century that even the errors showed a normal distribution pattern.

The normal curve was used to standardize the data sets as well as to analyze errors and patterns of the error distribution. For example, in astronomical observation measurements, the normal curve was used to analyze errors.

The similar distribution was found by the renowned French mathematician Laplace in the late 18th century. The central limit theorem of Laplace states that the sample means distribution follows the normal standard distribution and that the larger the data set the more the distribution deviates to the normal distribution.

Abraham de Moivre’s book entitled ‘Doctrine of Chances’ published another theorem in collaboration with the Central Limit Theorem. The ‘Moivre-Laplace’ theorem states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions.

Part 5: Practice out the Z-score calculation

Let’s practice out a few simple questions and find out the z-score. The formula for z-score calculation is

Z-score = (value – mean)/standard deviation

or

Z score = ( x – µ ) / σ

Question # 1
300 college student’s exam scores are tallied at the end of the semester. Eric scored 800 marks (X) in total out of 1000. The average score for the batch was 700 (µ) and the standard deviation was 180 (σ). Let’s find out how well Eric scored compared to his batch mates.

  1. -0.56
  2. 0.56
  3. -0.55
  4. 0.36

Question # 2

The score of students in the Statistics exam is normally distributed with μ=55 and σ=12. Zarine scored 65 on the exam. What is the z-score for George’s exam grade?

  1. 89
  2. 0.83
  3. 0.89
  4. 1.83

Question # 3

The height of the players in a football team is normally distributed with μ=68 inches and σ=5 inches. Xavier’s height is 61 inches. What is the z-score corresponding to his height?

  1. 1.4
  2. -2.15
  3. -1.4
  4. -1.5

Question # 4

A team conducted a survey of families residing in a small town. The survey revealed that the average family income is $35,000 and σ=$8,538. The Holmes family income is $67,000. Choose a corresponding z-score:

  1. 3.74
  2. -3.74
  3. 1.1
  4. 2.56

You may also calculate the z-score using z-score calculator.

What’s next?

To calculate the probability of above z-table:

  • Look at the value of the first two digits on the Y axis
  • Go to the X-axis to find a second decimal value

Read this article to understand how to use the z-table.