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Course: Digital SAT Math > Unit 11

Lesson 4: Center, spread, and shape of distributions: advanced
Test prep
Digital SAT Math
Advanced: Problem solving and data analysis
Center, spread, and shape of distributions: advanced
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Center, spread, and shape of distributions | Lesson

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A guide to center, spread, and shape of distributions on the digital SAT

What are center, spread, and shape of distributions?

Center, spread, and shape of distributions are also known as summary statistics (or statistics for short). These measurements are used to concisely describe data sets.
  • Center describes a typical value of in a data set. The SAT covers three measures of center: mean, median, and occasionally mode.
  • Spread describes the variation of the data. Two measures of spread are range and standard deviation.
You can learn anything. Let's do this!

What do the measures of center represent?

Statistics intro: mean, median, & mode

Khan Academy video wrapper
Statistics intro: Mean, median, & modeSee video transcript

How do I find the mean, median, and mode?

On the SAT, we need to know how to find the mean, median, and mode of a data set.

Mean

The mean is the average value of a data set.
mean=sum of valuesnumber of values
Example:
2, 5, 6, 7, 10
What is the mean of the data set above?
Example:
Pets ownedNumber of students
04
13
23
32
A teacher asked 12 students how many pets they owned. The results are shown in the table above. What is the average number of pets owned by the students?

Median

The median is the middle value when the data are ordered from least to greatest.
  • If the number of values is odd, the median is the middle value.
  • If the number of values is even, the median is the average of the two middle values.
Example:
9, 7, 12, 5, 9
What is the median of the data set above?
Example:
2, 5, 6, 7, 7, 10
What is the median of the data set above?

Mode

The mode is the value that appears most frequently in a data set. A data set can have no mode if no value appears more than any other; a data set can also have more than one mode.
Example:
1, 1, 2, 3, 3, 3, 3, 3, 8
What is the mode of the data set above?

Try it!

Try: find the centers of a distribution
ItemPrice (dollars)
VHS tape3
Salad bowl5
Salt box2
Hammock15
Concert poster5
Hoodie5
Raccoon statue7
The table above shows the items Stevie bought from a garage sale and their prices.
What is the mean price of the items Stevie bought?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
dollars
What is the median price of the items Stevie bought?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
dollars
What is the mode of the prices?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
dollars

What do the measures of spread represent?

Measures of spread: range, variance & standard deviation

Khan Academy video wrapper
Measures of spread: range, variance & standard deviationSee video transcript
Note: variance is not covered on the SAT, and while you may be asked about standard deviation, you will not need to calculate it on your own.

How do I find the range and standard deviation?

On the SAT, we need to know how to find the range of a data set. While we won't be asked to calculate the standard deviation, we do need to have a sense of the relative standard deviations of two data sets.

Range

The range measures the total spread of the data; it is the difference between the maximum and minimum values.
range=maximum value−minimum value
A larger range indicates a greater spread in the data.
Example:
1, 9, 4, 3, 8
What is the range of the data set above?

Standard deviation

Standard deviation measures the typical spread from the mean; it is the average distance between the mean and a value in the data set.
Larger standard deviations indicate greater spread in the data.
Example:
A number line is marked from 0 to 20 at intervals of 5. The dots are distributed as follows: 5, 2 dots; 10, 1 dot; 15, 2 dots.
A dotplot is marked from 0 to 20 at intervals of 5 units. There is one dot stacked above each marked value.
Of the two dot plots shown above, which one has a greater standard deviation?

Try it!

Try: compare two distributions
Guitar practice time in minutes
DayJazminPablo
Monday3030
Tuesday450
Wednesday3045
Thursday4530
Friday450
Saturday60120
Sunday6090
The table above shows the amount of time Jazmin and Pablo spent practicing guitar last week.
The range of Jazmin's practice times is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
minutes.
The range of Pablo's practice times is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
minutes.
Both Jazmin and Pablo practiced an average of 45 minutes a day. However, because Jazmin's practice times are
the 45-minute mean than Pablo's, the standard deviation of Jazmin's practice times is
that of Pablo's practice times.

How do outliers affect summary statistics?

Impact on median & mean: removing an outlier

Khan Academy video wrapper
Impact on median & mean: removing an outlierSee video transcript

The effect of outliers

An outlier is a value in a data set that significantly differs from other values. The inclusion of outliers in data sets can greatly skew the summary statistics, which is why outliers are often removed from data sets.

Effect on the range and standard deviation

The inclusion of outliers increases the spread of data, leading to larger range and standard deviation. Conversely, removing outliers decreases the spread of data, leading to smaller range and standard deviation.

Effect on the mean

An outlier can significantly skew the mean of a data set. For example, consider the data set {3,5,7,7,10,100}.
100 is an outlier; it is significantly larger than the other values in the data set. If we include the 100, the mean of the data set is:
3+5+7+7+10+1006=22
Notice that the mean, 22, is greater than 5 of the 6 values in the data set! If we remove the 100, however, the mean of the remaining values is:
3+5+7+7+105=6.4
The removal of an outlier is guaranteed to change the mean.
  • If a very large outlier is removed, the mean of the remaining values will decrease.
  • If a very small outlier is removed, the mean of the remaining values will increase.

Effect on the median

The median of the data set {3,5,7,7,10,100} is 7.
If we remove the outlier 100, the median of the remaining values, {3,5,7,7,10}, is still 7 !
Because the median is based on the middle values of a data set, an outlier does not affect the median of a data set as strongly as it affects the mean. As such, the removal of an outlier can still change the median, but that change is not guaranteed.
  • If a very large outlier is removed, the median of the remaining value will either decrease or remain the same.
  • If a very small outlier is removed, the median of the remaining value will either increase or remain the same.

Try it!

Try: determine the effect of removing an outlier
A dot plot represents height in inches. The horizontal axis is marked from 41 to 57 at intervals of 1. The dots are distributed over the values as follows: 42, 1 dot; 46, 1 dot; 47, 2 dots; 48, 1 dot; 49, 2 dots; 50, 2 dots; 51, 3 dots; 52, 2 dots; 53, 2 dots; 54, 2 dots; 55, 1 dot; 56, 1 dot.
The dot plot above shows the height in inches of 20 elementary school students.
If the shortest student is removed from the data set and the summary statistics are re-calculated, how would they compare to the summary statistics for all 20 students?
The mean height of the 19 remaining students would be
that of all 20 students.
The median height of the 19 remaining students would be
that of all 20 students.
The range of the heights of the 19 remaining students would be
that of all 20 students.

How do I use the mean to calculate a missing value?

Missing value given the mean

Khan Academy video wrapper
Missing value given the meanSee video transcript

How do I solve for a missing value?

If we know the mean of a data set and the number of values, we can calculate a missing value in the data set by:
  1. Calculating the sum of values by multiplying the mean by the number of values.
  2. Subtract all known values from the sum of values.
Example:
20, 20, 40, 60, x
If the mean of the five numbers above is 30, what is the value of x ?

Try it!

Try: find a missing value using the mean
GamePoints scored
111
2x
313
47
59
612
The table above shows the number of points Marco scored in the last six basketball games he played. Marco doesn't remember how many points he scored in game 2, but his coach tells him he averaged 10 points per game.
What is the total number of points Marco scored in the six games?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
points
How many points did Marco score in games 1, 3, 4, 5, and 6 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
points
How many points did Marco score in game 2 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
points

Your turn!

Practice: compare two distributions
NameTest 1Test 2Test 3Test 4Test 5
Amara9895949395
Lance96951008896
Amara and Lance are taking the same class. The table above shows their test scores for the class. Which of the following statements about their test scores is true?
Choose 1 answer:

Practice: find the median given frequency data
A histogram represents the number of acres that produce different soybean yields in bushels. Soybean yields range from 35 to 75, with classes of width 5. The following list provides the number of acres for each yield interval: 40 to 45 bushels, 25 acres; 45 to 50 bushels, 70 acres; 50 to 55 bushels, 55 acres; 55 to 60 bushels, 20 acres; 60 to 65 bushels, 5 acres; 65 to 70 bushels, 0 acres; 70 to 75 bushels, 0 acres.
Ned runs a soybean farm and recorded the yields for 175 different one-acre sections. The results are shown in the graph above. Which of the following could be the median yield of Ned's soybean acres?
Choose 1 answer:

Practice: determine the effects of changing a data set
The minimum value of a data set consisting of 15 positive integers is 29. A new data set consisting of 16 positive integers is created by including 22 in the original data set. Which of the following measures must be 7 greater for the new data set than for the original data set?
Choose 1 answer:

Practice: find a missing value using the mean
Last week, George drove an average of 52 miles per day. If the day he drove the longest distance is removed, the average distance he drove in the remaining 6 days becomes 40 miles per day. What was the longest distance, in miles, George drove in a single day last week?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Things to remember

mean=sum of valuesnumber of values
The median is the middle value when the data are ordered from least to greatest.
  • If the number of values is odd, the median is the middle value.
  • If the number of values is even, the median is the average of the two middle values.
The mode is the most common value in a data set.
range=maximum value−minimum value
Standard deviation measures the typical spread from the mean.

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