If a normal distribution shows that the height of adult women has a mean of 64.5 inches and a standard deviation of 2.5 inches, what does a z-score of -1.4 mean?
A z-score is a measure of how many standard deviations a particular data point is from the mean of a distribution.
In this case, a z-score of -1.4 indicates that a data point is 1.4 standard deviations below the mean.
To calculate the corresponding height, we will use the formula:
z = (x - μ) / σ
Rearranging the formula, we have:
x = μ + (z * σ)
Substituting the given values:
x = 64.5 + (-1.4 * 2.5)
x = 64.5 - 3.5
x = 61
Therefore, a z-score of -1.4 indicates that the height of an individual is 61 inches, which is 1.4 standard deviations below the mean height of adult women.
A z-score measures how many standard deviations an individual data point is from the mean of a distribution. In this case, a z-score of -1.4 means that the data point is 1.4 standard deviations below the mean.
To calculate the exact value represented by the z-score, we can use the formula:
z = (x - μ) / σ
Where:
z = z-score
x = data point
μ = mean
σ = standard deviation
In this case, the mean (μ) is 64.5 inches and the standard deviation (σ) is 2.5 inches. Plugging these values into the formula, we get:
-1.4 = (x - 64.5) / 2.5
To find the value of x, we can rearrange the formula:
(x - 64.5) / 2.5 = -1.4
Multiply both sides of the equation by 2.5:
x - 64.5 = -1.4 * 2.5
x - 64.5 = -3.5
Add 64.5 to both sides of the equation:
x = -3.5 + 64.5
x = 61
Therefore, a z-score of -1.4 corresponds to a height of approximately 61 inches.
To understand what the z-score of -1.4 means in this context, we need to understand what a z-score represents.
A z-score, also known as a standard score, is a measure of how many standard deviations an individual data point is from the mean of a distribution. It helps us understand where a particular data point stands in relation to the rest of the distribution.
In this case, the given normal distribution represents the heights of adult women, with a mean of 64.5 inches and a standard deviation of 2.5 inches.
To determine the meaning of a z-score of -1.4, we can use the formula:
z = (x - μ) / σ
Where:
z is the z-score,
x is the value we want to convert to a z-score,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Let's calculate the value corresponding to the z-score of -1.4:
z = -1.4
μ = 64.5 inches
σ = 2.5 inches
Rearranging the formula:
x = z * σ + μ
Substituting the values:
x = -1.4 * 2.5 + 64.5
x = -3.5 + 64.5
x = 61
Therefore, a z-score of -1.4 corresponds to a height of approximately 61 inches. This means that the height of an individual with a height of 61 inches is 1.4 standard deviations below the mean height of adult women.