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A Z-Score, also known as a standard score, is a statistical measurement that indicates how many standard deviations a particular data point is from the mean of a data set. It is commonly used to compare data points from different distributions.
In some cases, a Z-Score can be negative. This occurs when a data point falls below the mean of a data set. A negative Z-Score indicates that the data point is below the average and is considered to be a low value in comparison to the rest of the data set.
However, it is important to note that a negative Z-Score does not necessarily mean that the data point is bad or incorrect. It simply means that it is lower than the average and falls on the left side of the bell curve. Therefore, a negative Z-Score should not be automatically considered as a negative or undesirable value.
Can a Z-Score Be Negative?
In statistics, a z-score tells us how many standard deviations away a value is from . We use the following formula to calculate a z-score:
z = (X – μ) / σ
where X is the value we are analyzing, μ is the mean, and σ is the standard deviation.
A z-score can be positive, negative, or equal to zero.
A positive z-score indicates that a particular value is greater than the mean, a negative z-score indicates that a particular value is less than the mean, and a z-score of zero indicates that a particular value is equal to the mean.
A few examples should make this clear.
Examples: Calculating a Z-Score
Suppose we have the following dataset that shows the height (in inches) of a certain group of plants:
5, 7, 7, 8, 9, 10, 13, 17, 17, 18, 19, 19, 20
The sample mean of this dataset is 13 and the sample standard deviation is 5.51.
1. Find the z-score for the value “8” in this dataset.
Here is how to calculate the z-score:
z = (X – μ) / σ = (8 – 13) / 5.51 = -0.91
This means that the value “8” is 0.91 standard deviations below the mean.
2. Find the z-score for the value “13” in this dataset.
Here is how to calculate the z-score:
z = (X – μ) / σ = (13 – 13) / 5.46 = 0
3. Find the z-score for the value “20” in this dataset.
Here is how to calculate the z-score:
z = (X – μ) / σ = (20 – 13) / 5.46 = 1.28
This means that the value “20” is 1.28 standard deviations above the mean.
How to Interpret Z-Scores
A Z Table tells us what percentage of values fall below certain Z-scores. A few examples should make this clear.
Example 1: Negative Z-Scores
Earlier, we found that the raw value “8” in our dataset had a z-score of -0.91. According to the Z Table, 18.14% of values fall below this value.
Example 2: Z-Scores equal to zero
Earlier, we found that the raw value “13” in our dataset had a z-score of 0. According to the Z Table, 50.00% of values fall below this value.
Example 3: Positive Z-Scores
Earlier, we found that the raw value “20” in our dataset had a z-score of 1.28. According to the Z Table, 89.97% of values fall below this value.
Conclusion
Z-scores can take on any value between negative infinity and positive infinity, but most z-scores fall within 2 standard deviations of the mean. There’s actually a rule in statistics known as , which states that for a given dataset with a normal distribution:
- 68% of data values fall within one standard deviation of the mean.
- 95% of data values fall within two standard deviations of the mean.
- 99.7% of data values fall within three standard deviations of the mean.
The higher the absolute value of a z-score, the further away a raw value is from the mean of the dataset. The lower the absolute value of a z-score, the closer a raw value is to the mean of the dataset.
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