The first way（Given the mean value and standard tolerance）：
Test Value(Ⅹ)：
Mean Value(μ)：
Standard Deviation(σ)：
z score result
z = (Ⅹ - μ) / σ
The second way（Given dataset）：
Test Value(Ⅹ)：
Input Data Set:
The z-score, commonly known as the standard score, z-value, or normal score, is a dimensionless measure that indicates how many standard deviations an event is from the mean value being assessed. Events that fall above the mean are represented by positive z-scores, whereas those below the mean are indicated by negative z-scores. To calculate the z-score, one subtracts the population mean from the specific raw score or data point (such as a test score, height, age, etc.) and then divides this difference by the population standard deviation:\(z = \frac{X - \mu}{\sigma} \), In this context, X represents the raw score, \(\mu\) denotes the population mean, and \(\sigma\) indicates the population standard deviation. When dealing with a sample, the formula is quite similar, but it utilizes the sample mean and the sample standard deviation instead of the population parameters.
The z-score is a value utilized to characterize the normal distribution. It represents the distance between the mean score and a specific data point, expressed in terms of standard deviations (SD). In the context of statistical data analysis, it is also referred to as the standard score, z value, standardized score, or normal score. To calculate the z-score, you must first determine the mean and standard deviation of the data set. The mean, represented by the symbol μ, is calculated by summing all the values in the data set and dividing by the total number of data points. This can be expressed as \(\mu = \frac{\sum_{i=1}^n X_i}{n}\). The standard deviation is calculated using the following expression:: \(\sigma = \sqrt{\frac{\sum_{i=0}^n (X_i - μ)²}{n}}\) , In this context, xx represents the raw value, while nn denotes the total number of data points. To calculate the z-score, you simply need to use the following formula: \(z = \frac{X - \mu}{\sigma} \)
Let's assume the following task: during a test, four students scored 10, 12, 14, 16,and 18 points. What is the z-score of the result 16?
Find the mean of the results: \(\mu = \frac{10 + 12 + 14 + 16 + 18}{5} = 14\)
\((10 - 14)^2 = 16\)
\((12 - 14)^2 = 4\)
\((14 - 14)^2 = 0\)
\((16 - 14)^2 = 4\)
\((18 - 14)^2 = 16\)
Calculate the standard deviation: \(\sqrt{\frac{16 + 4 + 0 + 4 + 16}{5}} = \sqrt{\frac{40}{5}} = 2.83\)
Input these results to the z-score equation for x = 16: \(z = \frac{16 - 14}{2.83} = 0.71\)
Yes, a negative z-score indicates that your data point is lower than the mean!
The easiest way to find the p-value from the z-score is to use a z-score table. The actual calculation involves integrating the area under the curve of a normal distribution.
