- The table shows the mean and standard deviation of daily sales (in hundreds of dollars) for a store over four consecutive weeks. Based on the table, during which week were daily sales most consistent?
Answer: Week 3
Most consistent' means least variability relative to the mean. Compute coefficient of variation (SD/mean): Week 1: 6/30 = 0.20. Week 2: 4/20 = 0.20. Week 3: 3/40 = 0.075. Week 4: 5/25 = 0.20. Week 3 has by far the lowest relative variability, so sales were most consistent. A student who chooses (B) based on smallest absolute SD (4 in Week 2) misses that Week 3's SD of 3 is even smaller. A student choosing (D) may confuse small mean with small variability.
- Which of the following statements about the range and standard deviation is true?
Answer: Removing an outlier from a dataset will typically have a larger impact on the range than on the standard deviation.
The range depends only on the two most extreme values in a dataset. The standard deviation depends on every value in the dataset. Therefore, removing one of the extreme values (an outlier) will directly and significantly change the range, as a new maximum or minimum is established. While it will also affect the standard deviation, the impact is often less dramatic as the standard deviation is an average of all deviations. This makes the range more sensitive to single outliers than the standard deviation.
- A dataset consists of five distinct positive integers. If every number in the dataset is increased by 10, how does this transformation affect the range and standard deviation of the dataset?
Answer: The range and the standard deviation both remain the same.
Adding a constant value to every data point in a set shifts the entire set on the number line but does not change the spread of the data. The difference between the maximum and minimum values (the range) will be unchanged because both values are shifted by the same amount. Similarly, the standard deviation, which measures the dispersion around the mean, will also remain the same because each value's distance from the (new) mean is preserved.
- The standard deviation of a dataset is 2.5. A new dataset is created by multiplying each number in the original dataset by 4 and then adding 7. What is the standard deviation of the new dataset?
Answer: 10
This transformation involves two steps: multiplication and addition. Adding a constant (7) to every data point shifts the dataset but does not change its spread, so it has no effect on the standard deviation. Multiplying every data point by a constant (4) scales the spread of the data. The new standard deviation will be the original standard deviation multiplied by the absolute value of that constant. So, the new standard deviation is 2.5 * 4 = 10.
- A factory manager is comparing the performance of two machines that fill bags of flour. The target weight is 500 grams. The weights of bags from Machine X have a standard deviation of 2 grams. The weights of bags from Machine Y have a standard deviation of 5 grams. Which statement is best supported by this data?
Answer: The weights of bags filled by Machine X are more consistent than those from Machine Y.
A smaller standard deviation implies less variability and therefore greater consistency. Since Machine X has a standard deviation of 2 grams, which is smaller than Machine Y's 5 grams, the weights of the bags it produces are more consistent and closer to the average weight. Standard deviation does not provide information about the average (mean) weight, the speed of the machines, or the probability of any specific outcome.
- A researcher has a dataset of 100 measurements. The researcher removes the single highest value and the single lowest value from the dataset. Which of the following effects will this action most likely have on the range and standard deviation?
Answer: The range will decrease, and the standard deviation will likely decrease.
The range is calculated as the difference between the maximum and minimum values. By removing these two specific values, the new range will be calculated from less extreme values, so the range must decrease (unless all values were identical). The standard deviation measures the overall spread from the mean. Removing the two most extreme values (outliers) will almost always reduce the overall variability of the data, causing the standard deviation to decrease as well.
- The scatter plot shown displays the relationship between study hours and test scores for 20 students. Based on the scatter plot shown, which of the following best describes the variability in test scores?
Answer: The variability in test scores decreases as study hours increase.
Reading the scatter plot: at low study hours (1–2 hours), scores range widely from about 40 to 85 (high variability). At high study hours (7–8 hours), scores cluster tightly from about 85 to 95 (low variability). This pattern — decreasing spread as the x-variable increases — indicates decreasing variability, a concept called heteroscedasticity. (A) ignores the visible funnel pattern. (B) reverses the direction. (D) is incorrect because variability clearly exists, especially at low study hours.
- The following dataset represents the number of daily visitors to a small museum over a week: {50, 55, 48, 52, 58, 60, 51}. If on the eighth day, there was a special event and the museum received 150 visitors, how would the inclusion of this eighth day's data affect the variability of the dataset?
Answer: Both the range and the standard deviation would increase.
The original data is tightly clustered. The value of 150 is an outlier, as it is significantly higher than the other data points. Adding an outlier to a dataset increases its overall spread. The range (the difference between the maximum and minimum values) will increase because the maximum value is now much larger. The standard deviation, which measures the average distance of data points from the mean, will also increase because the new point is very far from the original mean.
- A researcher is studying the weights of apples from two different orchards. The weights from Orchard F are described as being very uniform, with most apples having weights very close to the average. The weights from Orchard G vary significantly, with some apples being quite small and others quite large. Which of the following statements is most likely true?
Answer: The standard deviation of weights from Orchard G is greater than that from Orchard F.
The description indicates that the weights from Orchard F have low variability ('very uniform'), while the weights from Orchard G have high variability ('vary significantly'). Standard deviation is a measure of this variability. Therefore, it is most likely that the standard deviation for the more variable set (Orchard G) is greater than the standard deviation for the less variable set (Orchard F). The information given does not allow for any conclusions about the mean weights.
- Which of the following datasets has the greatest standard deviation?
Answer: {0, 0, 20, 40, 40}
Without calculating, we can compare the spread conceptually. All sets except B have a mean of 20. Set B has a standard deviation of 0 because all values are the same. Set D has values tightly clustered around 20. Set A has values evenly spaced. Set C has values clustered at the extremes (0 and 40), which are furthest from the mean of 20. This polarization of data points results in the largest standard deviation.
- A teacher grades two different classes on the same 20-point quiz. The scores for Class A have a mean of 15 and a standard deviation of 1.5. The scores for Class B have a mean of 15 and a standard deviation of 3.2. Which of the following is the most accurate conclusion that can be drawn from this information?
Answer: The scores in Class A are more clustered around the mean than the scores in Class B.
Standard deviation is a measure of the spread or dispersion of a set of data from its mean. A smaller standard deviation (1.5 for Class A) indicates that the data points tend to be very close to the mean. A larger standard deviation (3.2 for Class B) indicates that the data points are spread out over a wider range of values. Therefore, the scores in Class A are more clustered around the mean. The other choices make assumptions that cannot be supported by the given information; standard deviation does not determine the highest score or the likelihood of any single score.
- The histogram shown displays the distribution of reaction times (in milliseconds) for 50 participants in a study. If a second group of 50 participants had the same mean reaction time but smaller standard deviation, which of the following would best describe how their histogram would compare to the one shown?
Answer: The bars would be more concentrated near the center, with shorter bars at the tails.
A smaller standard deviation with the same mean indicates data is more tightly clustered around the center. The histogram would show taller bars near the mean and shorter (or empty) bars at the tails. (A) describes a change in mean, not variability. (C) would reduce the total count. (D) describes a change in symmetry/skewness, not spread.
- Four archers each shoot six arrows at a target. Archer 1's arrows are all in the bullseye. Archer 2's arrows are spread widely all over the target. Archer 3's arrows are tightly clustered, but in the top-left corner of the target. Archer 4's arrows form a straight vertical line through the center of the target. Which archer's shots have the smallest standard deviation in their distances from the center of the bullseye?
Answer: Archer 1
Standard deviation measures the spread of data points from a central value (in this case, the center of the bullseye). Archer 1's arrows are all in the bullseye, meaning their distance from the center is zero. A dataset with all identical values has a standard deviation of 0, which is the smallest possible value. Archer 3's shots have a small standard deviation relative to their own average position, but a large standard deviation relative to the bullseye. Archer 2 and 4 have shots with clear and significant spread from the center.
- The box plots compare the daily high temperatures (in °F) recorded in two cities during the month of June. Based on the box plots shown, which of the following must be true?
Answer: The interquartile range of City X is greater than the interquartile range of City Y.
The IQR is the length of the box (Q3 − Q1). City X's box spans 70–85 (IQR = 15), while City Y's box spans 75–82 (IQR = 7), so City X has a greater IQR. (A) is wrong because both medians are at 78. (C) is wrong because both have the same total range (60 to 95). (D) cannot be determined from box plots — box plots show medians, not means.
- The monthly rainfall, in inches, for two cities was recorded for a year. For City A, the data has a range of 4 inches. For City B, the data has a range of 9 inches. Based solely on this information, what can be inferred about the standard deviation of the monthly rainfall for the two cities?
Answer: The standard deviation for City B is likely greater than for City A.
The range is a measure of variability, calculated as the difference between the maximum and minimum values. Standard deviation is another measure of variability that considers the spread of all data points around the mean. While they are different measures, a significantly larger range (9 inches for City B vs. 4 inches for City A) strongly suggests that the data for City B is more spread out, and thus its standard deviation is also likely to be greater. One cannot determine the mean or the exact standard deviation from the range alone.
- The cumulative frequency graph shown represents the exam scores of 200 students. Based on the cumulative frequency graph, what is the interquartile range of the exam scores?
Answer: 30
IQR = Q3 − Q1. Q1 corresponds to cumulative frequency 50 (25% of 200), and Q3 corresponds to cumulative frequency 150 (75% of 200). Reading the graph: at cumulative frequency 50, score ≈ 60; at cumulative frequency 150, score ≈ 90. IQR = 90 − 60 = 30. (A) 20 likely comes from misreading quartile positions. (C) 50 is the median value, not IQR. (D) 70 is the full range, not IQR.
- A dataset has a very small standard deviation, but its mean is a large positive number. Which of the following best describes the data points in the set?
Answer: The data points are all large positive numbers and are located close to one another.
The mean being a large positive number indicates the center of the data is a large positive number. The standard deviation being very small indicates that the data points have very little spread and are tightly clustered around the mean. Combining these two facts, we can conclude that the data points are all large positive numbers and are located close to one another.
- A quality inspector found that a batch of resistors had a mean resistance of 200 ohms with a standard deviation of 0. Which of the following statements must be true about the batch of resistors?
Answer: Every resistor in the batch has a resistance of exactly 200 ohms.
The standard deviation is a measure of the spread of data. A standard deviation of 0 means there is no spread or variability in the data whatsoever. For this to be true, all data points in the set must be identical to the mean. Therefore, every resistor in the batch must have a resistance of exactly 200 ohms.
- The number of defective parts per 1,000 produced by a factory is recorded each day for ten days. The dataset is {2, 3, 2, 4, 3, 1, 2, 3, 8, 2}. The manager removes the data point '8' because it was caused by a machine malfunction that has since been fixed. How will removing this data point affect the mean and standard deviation of the dataset?
Answer: The mean will decrease, and the standard deviation will decrease.
The data point '8' is an outlier, as it is significantly higher than the other values. Removing a value that is above the original mean will cause the new mean to decrease. Removing an outlier also reduces the overall spread or variability of the data. Therefore, the standard deviation will also decrease.
- The two data sets shown in the table below have the same number of values. Based on the table, which of the following correctly compares the standard deviations of Data Set P and Data Set Q?
Answer: The standard deviation of P is less than the standard deviation of Q because P's values are closer to its mean.
Both sets have mean 20 and range 20. However, Data Set P's values (10, 19, 20, 21, 30) cluster tightly around 20 except for extremes, while Data Set Q's values (10, 15, 20, 25, 30) are spread uniformly. The deviations from the mean in Q are larger on average, so Q has a greater standard deviation. (A) uses only the maximum, which is irrelevant. (C) and (D) are common misconceptions — equal mean or equal range does not imply equal standard deviation.
