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### Course: Digital SAT Math > Unit 3

Lesson 4: Center, spread, and shape of distributions: foundations# Center, spread, and shape of distributions | Lesson

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

### 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.

**Example:**

What is the mean of the data set above?

**Example:**

Pets owned | Number of students |
---|---|

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:**

What is the median of the data set above?

**Example:**

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:**

What is the mode of the data set above?

### Try it!

## What do the measures of spread represent?

### Measures of spread: range, variance & standard deviation

**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.A larger range indicates a greater spread in the data.

**Example:**

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:**

Of the two dot plots shown above, which one has a greater standard deviation?

### Try it!

## How do outliers affect summary statistics?

### Impact on median & mean: removing an outlier

### 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 $\{3,5,7,7,10,100\}$ .

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:

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!

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

### Missing value given the mean

### 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:

- Calculating the sum of values by multiplying the mean by the number of values.
- Subtract all known values from the sum of values.

**Example:**

If the mean of the five numbers above is $30$ , what is the value of $x$ ?

### Try it!

## Your turn!

## Things to remember

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.

Standard deviation measures the

*typical spread*from the mean.## Want to join the conversation?

- My brain is not braining anymore *-*(209 votes)
- sounds like me :)(31 votes)

- can't be the only one who thinks the 7 range question is phrased horridly(123 votes)
- not at all, its a very straightforward question 💀(0 votes)

- didnt uderstand that how come 48 us the answer for the median of the yield.(46 votes)
- I suggest you watch this video by Sal for a better understanding

https://www.khanacademy.org/math/ap-statistics/summarizing-quantitative-data-ap/measuring-center-quantitative/v/median-histogram(57 votes)

- Guys its tomorrow for me aaaaaaaaaaa (pls pray for me n you)(42 votes)
- how did you do bro?(7 votes)

- standard deviation is confusing 😭(31 votes)
- bro some stuff is confusing esp from the soybean one until the end(29 votes)
- What exactly is a standard deviation??(6 votes)
- The standard deviation is the average amount of variability in your dataset. It tells you, on average, how far each value lies from the mean. A high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean. So it basically tells you how close on average the numbers in your data set are to the mean of your data set. If the standard deviation is low, that means the numbers in your data set are clustered around the mean, whereas if the standard deviation is high, it means the numbers are more spread out from the mean. I hope this answer makes sense!(35 votes)

- in the practice section, i cant understand that bar graph question(8 votes)
- The number of one acre sections taken is 175. The median of that is the [(175+1)/2]= 88th section. Since the ranges of acres are arranged in ascending order, we can use the graph as is. We see that 40-45 bushels occur in 25 sections- not enough. Next, 45-50 bushels occur in the next 70 sections- the 88th section falls within this range. Since 48 bushels is the only option that falls into the corresponding bushel range, that is our answer.(19 votes)

- (For Soybeans one)

On the graph:

the x-axis represents the yield of soybeans in bushels from the soybean acres

the y-axis represents the number of acres

The average of the 175 acre section is 175/2 which is 87.5 or approximated to 88

Using the number of acres from the first 2 bars: 0-25 and 0-70

25 + 70 = 95 acres

95 acres is greater than 88. Thus, the 88th acre will be found within the 2nd bar where the yields of soybeans are between 45 and 50 bushels. The answer should be greater than 45 and less than 50 (45 > x > 50).

From the options, the only answer that fits this criteria is 48 bushels(11 votes) - Why is the solution to the 3 problems before the last problem not explained? Normally after every math problem, there is 2 button called "Check And Solve" But I can not see any solve button for these 3 math! :((7 votes)