the distribution of scores on a test is mound shaped and dsymmetric with a mean score of 78. if 68 percent of the scores fall between 72 and 84, what is the stand deviaition
Approximately 68% of scores occur between mean ± one standard deviation. Does that help?
the answer is the standard deviation is 6 but i don't know how to get that
To find the standard deviation, we need to understand the concept of z-scores and the empirical rule.
1. Z-scores: A z-score measures the number of standard deviations a particular value is from the mean. Mathematically, the z-score is calculated as:
z = (x - μ) / σ
where z is the z-score, x is the individual value, μ is the mean, and σ is the standard deviation.
2. Empirical Rule: According to the empirical rule, for a mound-shaped or bell-shaped distribution:
- Approximately 68% of the scores fall within 1 standard deviation of the mean.
- Approximately 95% of the scores fall within 2 standard deviations of the mean.
- Approximately 99.7% of the scores fall within 3 standard deviations of the mean.
Based on the given information, 68% of the scores fall between 72 and 84, which is within 1 standard deviation of the mean. Since the mean is given as 78, we can set up the following equation to find the standard deviation:
72 = 78 - σ (lower end of the 1 standard deviation range)
84 = 78 + σ (upper end of the 1 standard deviation range)
Now, let's solve these equations to find the standard deviation.
78 - σ + σ = 72 + σ
78 = 72 + 2σ
6 = 2σ
σ = 6 / 2
σ = 3
Therefore, the standard deviation is 3.