The standard normal curve is symmetric about the value _
a) 0.5
b) 1
c) 8
d) 0
The correct answer and explanation is :
The correct answer is d) 0.
The standard normal distribution, also known as the z-distribution, is a special case of the normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. This distribution is symmetric about its mean value, which in this case is 0. This symmetry implies that the distribution is a mirror image on either side of the mean.
In a standard normal distribution, the mean, median, and mode all coincide at the value of 0. This central point divides the curve into two equal halves, each representing 50% of the data. The symmetry about the mean indicates that for any given positive deviation from the mean, there is an equally probable negative deviation of the same magnitude. This property is fundamental to the normal distribution and is utilized in various statistical analyses and hypothesis testing.
The bell-shaped curve of the standard normal distribution extends infinitely in both directions along the horizontal axis but approaches the axis asymptotically, meaning it gets closer and closer to the axis but never actually touches it. The total area under the curve is equal to 1, representing the entirety of possible outcomes for the random variable.
Understanding the symmetry of the standard normal curve is crucial when working with z-scores, which are standardized scores indicating how many standard deviations an element is from the mean. For example, a z-score of +1 indicates a value that is one standard deviation above the mean, while a z-score of -1 indicates a value one standard deviation below the mean. Due to the symmetry of the distribution, the probability associated with a z-score of +1 is equal to the probability associated with a z-score of -1.
This symmetry also simplifies the calculation of probabilities and critical values in statistical procedures. For instance, the probability of a value falling between the mean and one standard deviation above the mean is the same as the probability of it falling between the mean and one standard deviation below the mean. This property is often referred to when applying the empirical rule, which states that approximately 68% of data within a normal distribution lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
In summary, the standard normal curve is symmetric about the value 0, which is the mean of the distribution. This symmetry is a key characteristic that facilitates various statistical analyses and interpretations.