For an exam given to a class, the students' scores ranged from 34 to 99 , with a mean of 78 . Which of the following is the most realistic value for the standard deviation: -14,3,0,56,15? Clearly explain what's unrealistic about each of the other values.

The most realistic value for the standard deviation is 15.

Step-by-step explanation:

The standard deviation of a distribution is a measure of dispersion. It is a measure of the spread of the distribution from the mean of the distribution. It expresses how far most of the distribution is from the mean.

Mathematically, the standard deviation is given as the square root of variance. And variance is an average of the squared deviations from the mean.

Mathematically,

Standard deviation = σ = √[Σ(x - xbar)²/N]

x = each variable (ranges from 34 to 99)

xbar = mean = 78

N = number of variables

Now taking the given possible values of the standard deviation one at a time,

-14

The standard deviation cannot be negative as it is a square root of the average of the sum of square deviations from the mean. Since the square of a number cannot be negative, it directly translates that the standard deviation cannot be negative.

3

A small standard deviation like 3 indicates that the distribution mostly centres about the mean, with very little variation. And the distribution given has a mean (78) that is very far away from at least one of the variables in the distribution. Hence, 3 is too low to pass ad the standard deviation of this distribution described.

0

A standard deviation of 0 indicates that all the variables in the distribution have the same value as the mean. That is, the distribution only contains 1 number, probably multiple times. So, this cannot be the standard deviation for the distribution described.

56

This value represents a value that is too high to express the spread of the distribution described. The mean (78) is very close to the maximum value of the distribution, and far away from the lower value(s), indicating that most of the distribution is in and around the upper values with a few variables closer to the lower limit. A standard deviation as high as 56 for a mean of 78 translates to a distribution with most of variables far from the mean, which isn't the case here.

Moreso, a simple add of the standard deviation to the mean or subtracting the standard deviation from the mean should give at least one of the results with values within the distribution.

(Mean) + (Standard deviation) = 78 + 56 = 134 >> 99 (outside distribution)

(Mean) + (Standard deviation) = 78 - 56 = 22 << 34 (also outside the distribution)

15

This is the most realistic value for the standard deviation as it represents what the distribution described above is.

The mean (78) being close to the maximum value of the distribution, and far away from the lower value(s) indicates that most of the distribution is in and around the upper values with a few variables closer to the lower limit.

So, 15 indicates a perfect blend of small deviations due to the high values close to the mean and the very high deviation from the evidently few lower values.

(Mean) + (Standard deviation) = 78 + 15 = 93 < 99 (within distribution)

(Mean) + (Standard deviation) = 78 - 15 = 63 > 34 (also within the distribution)

Hope this Helps!!!