Z Score Table
Random variables are classified as discrete and continuous random variables. While the discrete variables can assume only countable number values, a continuous random variable can assume values only as intervals between two values. Their distribution curves are bell shaped and termed as approximately normal. The probability of a continuous random variable assuming a value in the given interval is proportional to the area of the distribution in the given interval.
Theoretically the normal distribution curve can be used to study many variables that are approximately normal.
The normal distribution function has the equation
$f(x)= \frac{\frac{e^{-(x- \mu)^{2}}}{2 \sigma^{2}}}{\sqrt[\sigma]{2 \pi}}$
Even though this equation is rather intimidating, in actual practice tables or technology is used for specific problems rather than the function itself. The formula tells the two parameters that determine the function are the mean of the distribution indicated by the Greek letter $‘ \mu’$ and the standard devation of the distribution denoted by $‘ \sigma’$. The following diagrams provide a comparison of the shape and position of two normal curves as related to these parameters.
Find below Z-score chart which can be used as a reference.
We will learn more about Z - score table with some good examples. You can also find help from an expert tutor, who can help you understand the basics.
In statistics, a data is analyzed for various study. A standard normal distribution means the items in the data are normally distributed. In all such cases, the mean is taken as 0 and the standard deviation is taken as 1.
In case of normal distribution, it is possible to convert any value to standard normal distribution table by knowing how many standard deviations is the number from mean of the data. The number of standard deviations from the mean is defined as the z score and is calculated by the formula
z = $\frac{(x – m)}{S}$, where ‘x’ is the selected item, ‘m’ is the mean and 'S' is the standard deviation.
Statisticians worked out and formed a table of z scores for different standard deviations. This table is called ‘z score table and it is very helpful to find more information on the data. The z score table also plays an important role in determining probability details.
The following bell shaped figure describes a normal distribution of a data.
It is seen that 68% of the area of the entire curve is covered within one standard deviation. It implies that, 68% of the data is covered within 1 standard deviation from the mean. Similarly, it can be seen that about 95% of the data is covered within 2 standard deviations from the mean. Thus it gives rise to the fact that the z score value is nothing but the fraction of the data for that particular z value.
Statisticians meticulously worked out the fractional data for decimals of z scores and formed the following z score table.
Example 1:
The test scores of students in a class test has a mean of 70 and with a standard deviation of 12. What is the probable percentage of students scored more than 85?
The z score for the given data is,
z = (85 – 70)/12 = 1.25
From the z score table the fraction of the data within this z score is 0.8944.
This means 89.44% of the students are within the test scores of 85 and hence the percentage oof students who are above the test score of 85 = (100 – 89.44)% = 10.56%
Hence, the required probable percentage is 10.56%.
Example 2:
An organization made a survey on the monthly salary of their clerical level employees, in dollars. The data revealed the mean as 4000 with a standard deviation of $600. Find what percentage of employees are in the salary bracket [3000, 4500].
The z score of the employees with a salary less than 3000 = (3000 - 4000)/600 = - 1.67 (approx)
The z score of the employees with a salary more than 4500 is, = (4500 - 4000)/600 = 0.83 (approx)
From the z score table, the fraction of the data within,
z score of -1.67 = 0.0475
z score of 0.83 = 0.7967
Therefore, the fraction of data between the z scores of -1.67 and 0.83 = 0.7967 – 0.0475 = 0.7492
Hence, 74.92% of clerical level employees are within the salary bracket [3000, 4500].
It was noted that tables can be used to find the areas representing the probabilities for normal distribution. Since the position and shape of normal curves vary, this would mean we require a table for each variable. To simplify this situation, the statisticians use Standardized normal distribution, and hence the Z –score table. If x denotes the normal variable used in the problem then z-score represents the corresponding variable in the standardized distribution.
The mean and standard deviation of a standardized normal distribution are 0 and 1.
The formula used for transformation into Z-score is
$Z = \frac {Value - Mean} { Standard Deviation} = \frac { x - \mu} {\sigma}$ Where $‘ \mu’$ and $‘ \sigma’$ are parameters of the original distribution.
Z score is the number of standard deviations the variable X is away from $ \mu $ .
The Z – table shows the area under the normal distribution curve, related to a Z-score.
1. Calculate the Z-score using the formula, $z = \frac {x- \mu} {\sigma}$ round the answer to the hundredth (two decimals)
2. Note the absolute value, ignoring the sign.
3. Read the Z value with the first decimal on left most column and move along the row to match the column showing the value in the second decimal place
4. Note down the area given inside the table.
Example:
The z score till the first decimal is $2.3$, read $2.3$ on the extreme left column of the table move 4 columns along the row counting $0.00, 0.01,0.02\ and\ 0.03$. Read the area given in the cell, which is $0.4901$.
Hence the area corresponding to $z = 2.33, = 0.4901$.
Due to symmetry the area corresponding to $z =-2.33\ is\ also\ 0.4901$.
$Z-Score \to Area\ Reading\ in\ Z-score\ table \to Probability Required$
We saw the steps of finding the area represented in the Z-score table corresponding to a z score. The procedure for determining area of the interval is explained aided with diagrams.
