To calculate the range, subtract the smallest value from the largest in the data set. For standard deviation and variance, find the mean first and then use the sum of the squared differences from the mean, divided by the number of samples minus one. These calculations tend to require a calculator.
To find the range, standard deviation, and variance for the given samples of 56, 93, 83, 46, 70, 55, 78, 53, and 97, we first arrange the data in ascending order: 46, 53, 55, 56, 70, 78, 83, 93, 97. The range is the difference between the largest and smallest values: 97 - 46 = 51.
To calculate the standard deviation and variance, we use the formula for the sample standard deviation and variance since we have a sample, not a population. The mean (μ) of our sample is calculated by adding all the numbers together and dividing by the number of samples: (μ) = (56 + 93 + 83 + 46 + 70 + 55 + 78 + 53 + 97) / 9 = 631 / 9 = 70.1. Next, we find the sum of the squared differences from the mean: Σ(xi - μ)^2 = (56-70.1)^2 + (93-70.1)^2 + … + (97-70.1)^2. Finally, we divide by the number of samples minus one to get the variance, and take the square root of the variance to get the standard deviation. These computations are typically done using a calculator due to the tedious nature of the calculations.
So, to find the value that is one standard deviation below the mean, subtract the standard deviation from the mean of the sample.