SAT · Problem-Solving and Data Analysis · TOPIC LESSON
Describe a distribution using its shape, center and spread without losing the raw-data meaning.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
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01 / UNDERSTAND & APPLY
One-variable data contain one measurement or category for each observation. A distribution shows the possible values and their frequencies. Before comparing displays, check the variable, units, sample sizes, axis scales and whether bars show counts or relative frequencies.
A frequency table with value 3 appearing 4 times contributes 12 to the total sum and 4 to the sample size. Do not average the distinct labels unless their frequencies are equal.
02 / UNDERSTAND & APPLY
A dot plot places one dot per observation and preserves individual values. A histogram groups numerical observations into intervals; its bars show frequencies or relative frequencies for bins. Exact individual values within a bin are normally unknown, so a histogram may support a range for a mean rather than an exact mean.
A bar chart for categories is different from a histogram for numeric intervals. For a histogram, read bin boundaries and endpoint conventions; a value on an edge must belong to exactly one bin. For unequal bin widths, check whether vertical height represents frequency or density before comparing areas.
03 / UNDERSTAND & APPLY
The arithmetic mean is sum of all observations divided by the number of observations. A weighted mean uses frequency × value in the numerator and total frequency in the denominator. To combine groups, add their totals, not their unweighted means unless their sizes match.
ORIGINAL WORKED EXAMPLE
A table lists value 2 with frequency 3, value 5 with frequency 2, and value 8 with frequency 1. Find the mean.
Answer 4
ORIGINAL WORKED EXAMPLE
Four measurements have mean 12 and six measurements have mean 17. What is the combined mean?
Answer 15
04 / UNDERSTAND & APPLY
Sort the observations first. With an odd count, the median is the middle value. With an even count, it is the average of the two middle values. In a frequency table, cumulative counts locate those positions. The median splits the ordered data by position; it need not be one of the observed values.
The mean uses every magnitude and is sensitive to extreme values. The median depends mainly on position and is more resistant, although adding or removing observations can still change it.
ORIGINAL WORKED EXAMPLE
Find the median of 9, 2, 7, 2, 12, 6.
Answer 6.5
05 / UNDERSTAND & APPLY
Range = maximum − minimum; it is easy to compute but depends only on two values. Standard deviation summarizes distances from the mean: squaring prevents positive and negative deviations from canceling, averaging summarizes them, and the square root restores the original units. A larger standard deviation means more spread about the mean.
You can often compare standard deviations without calculating them. Translating every value by the same constant changes the mean but not any distance from the mean. Multiplying every value by c multiplies the standard deviation by |c|. Population and sample formulas have different denominators; use the stated version if a numerical calculation is required.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A is 4,4,5,6,6 and B is 0,2,5,8,10. Which has larger standard deviation?
Answer B has the larger standard deviation
06 / UNDERSTAND & APPLY
A box plot shows a median and quartiles. Q1 marks the lower-quarter boundary and Q3 the upper-quarter boundary; IQR = Q3 − Q1 describes the middle half of observations. Whiskers may extend to minimum/maximum or to specified nonoutlier endpoints, depending on the convention. Read any legend before treating them as extremes.
The length of a box segment is not the number of observations in it: quartile portions contain roughly equal counts but may cover different numerical spans. For small datasets, quartile calculation conventions vary, so use supplied quartiles or the stated method.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
07 / UNDERSTAND & APPLY
An unusually high observation tends to pull the mean upward more than the median and often increases spread. A long right tail commonly places the mean above the median, but inspect the actual data instead of treating a visual rule as a universal theorem. Removing or adding an observation changes both the sum and the count.
Outliers are not automatically errors. Investigate context before excluding them. A single unusual value may be real and important.
ORIGINAL WORKED EXAMPLE
The values are 3,4,5,6,7. Add 35. Compare mean and median before and after.
Answer Mean: 5 to 10; median: 5 to 5.5
08 / UNDERSTAND & APPLY
Suppose a histogram has 3 observations in [0,10), 5 in [10,20), and 2 in [20,30). There are 10 observations total; 7 are at least 10. The fifth and sixth ordered observations are both in [10,20), so the median lies in that interval. The exact mean cannot be determined without the within-bin values.
A calculator can compute statistics from a frequency list, but ensure the frequency column is used and distinguish a population standard deviation from a sample standard deviation. A display can be misleading when axes or bin choices change between groups.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
09 / UNDERSTAND & APPLY
Use exact data when supplied; avoid inventing values within bins.
ORIGINAL PRACTICE · 1
5
The sum is 15 and there are 3 values.
ORIGINAL PRACTICE · 2
4.5
Sorted data are 1,3,6,8; average middle values 3 and 6.
ORIGINAL PRACTICE · 3
14
Maximum 12 minus minimum −2 gives 14.
ORIGINAL PRACTICE · 4
Mean increases by 9; standard deviation stays the same
Distances from the translated mean are unchanged.
ORIGINAL PRACTICE · 5
It is multiplied by 3
All distances from the mean triple.
ORIGINAL PRACTICE · 6
9
IQR is 21 − 12.
ORIGINAL PRACTICE · 7
2.8
The weighted total is 2 + 12 = 14 over 5 observations.
ORIGINAL PRACTICE · 8
No
Within-bin values are unknown; bin midpoints would provide an approximation rather than an exact mean.
ORIGINAL PRACTICE · 9
9
The old total is 80; the new total is 99 over 11 observations.
KEEP THE BIG IDEAS
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