3. Describing Data Numerically

Suppose you have two histograms, $A$ and $B$, each representing the distribution of exam scores for two different classes. Histogram $A$ shows scores tightly clustered around the mean, while Histogram $B$ shows scores spread out over a wider range. Which histogram depicts a higher $\mathrm{standard\; deviation}$?

Given the data set $1,2,4,5$, what is the variance of the data?

Suppose the weights of a group of students are normally distributed with a mean of $\mu =70 \mathrm{kg}$ and a standard deviation of $\sigma =5 \mathrm{kg}$. Which of the following weights are within $2$ standard deviations of the mean? Select three options.

If the sample variance of hourly wages is $10$, what is the sample standard deviation?

According to the range rule of thumb, the standard deviation of a data set can be roughly estimated as $\frac{\mathrm{Range}}{4}$.

Larger values of the standard deviation result in a normal curve that is: $\sigma$

In the calculation of a $z$-score, what does the denominator of the test statistic measure?

According to the empirical rule, what proportion of bond mutual funds' returns are expected to fall within $1$ standard deviation of the mean if the returns are approximately normally distributed?

If a distribution has zero variance, which of the following statements is true about the data values? $\mathrm{Var}\left(X\right)=0$

Finding Standard Deviation from a Frequency Distribution. In Exercises 37–40, refer to the frequency distribution in the given exercise and compute the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 18.5 minutes; (Exercise 38) 36.7 minutes; (Exercise 39) 6.9 years; (Exercise 40) 20.4 seconds.

Standard deviation for frequency distribution

According to the empirical rule, what proportion of bond mutual funds are expected to have returns within $1^{\sigma }$ of the mean if the returns are approximately normally distributed?

Suppose you have two histograms, $A$ and $B$, each representing the distribution of exam scores for two different classes. Histogram $A$ shows scores tightly clustered around the mean, while Histogram $B$ shows scores spread out over a wider range. Which histogram depicts a higher standard deviation?

In statistics, does the value of the $\mathrm{standard\; deviation}$ of a data set depend on the value of the $\mathrm{mean}$ of that data set?

Why is the $standard deviation$ generally considered a better measure of variation than the $range$?