Analyzing Data
What type of variable represents numerical data where arithmetic operations make sense?
Quantitative variable.
What does SOCS stand for when describing a distribution?
Shape, Outliers, Center, Spread (plus context).
What does a percentile represent?
The percentage of observations below a given value.
What is the area under the entire normal curve equal to?
1 (or 100%).
What type of graph is used to display two quantitative variables?
A scatterplot.
What is the difference between a response variable and an explanatory variable?
The explanatory variable helps explain or predict changes in the response variable.
Which measure of center is resistant to outliers: mean or median?
Median.
What is a z-score, and what does it tell you?
z = (x − mean)/SD; tells how many SDs an observation is from the mean.
A normal distribution has mean = 50 and SD = 10. What z-score corresponds to a value of 70?
z = (70 − 50)/10 = 2.
In a regression equation y-hat = a·x + b, what does the slope represent?
The predicted change in y for each unit increase in x.
Which graph would you use to display the relationship between two categorical variables?
Mosaic plot or segmented bar graph.
How do you find outliers using the 1.5×IQR rule?
Values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.
How does adding a constant affect the mean and standard deviation?
Changes the mean but not the standard deviation.
If a student’s z-score on a test is −1.5, what does that tell you about their performance?
The student scored 1.5 SDs below the mean.
If correlation (r) = 0, what does that indicate about the relationship between x and y?
There is no linear relationship between them.
What is the main advantage of a dot plot or stem-and-leaf plot over a histogram?
They display actual data values, not just frequencies.
If the data are skewed right, how do the mean and median compare?
Mean > Median.
How does multiplying data by a constant affect the mean and standard deviation?
Multiplies both the mean and SD by that constant.
Approximately what percent of observations fall between z = −2 and z = 1?
About 81.5% (from 2 SDs below to 1 SD above the mean).
How does an influential outlier affect the regression line?
It can significantly change the slope, y-intercept, and correlation coefficient (r) of the line.
How might a bar graph be misleading?
if the y-axis doesn’t start at zero
Two data sets have the same mean but different standard deviations. What does this tell you about how the data are distributed?
It means both sets are centered at the same value, but the one with the larger standard deviation has more variability — its data are more spread out around the mean.
What is a relative frequency graph used for?
Showing proportions or percentages instead of counts.
The distribution of IQ scores is normal with mean 100 and SD 15. What IQ corresponds to the 90th percentile?
z ≈ 1.28 → IQ = 100 + (1.28)(15) ≈ 119.2.
The correlation between study hours (x) and exam score (y) is 0.8. If SDₓ = 2 and SDᵧ = 10, what is the slope of the regression line?
a = r × (sᵧ/sₓ) = 0.8 × (10/2) = 4.0.