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Statistics and probability

Course: Statistics and probability > Unit 4

Lesson 3: Effects of linear transformations
Math
Statistics and probability
Modeling data distributions
Effects of linear transformations
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Transforming data problem

Google Classroom
It is very common to take data and apply the same transformation to every data point in the set. For example, we may take a set of temperatures taken in degrees fahrenheit and convert them all to degrees celsius. How would this conversion impact the measures of center of spread in the data set? Let's look at a simpler example to think about this situation.

Part 1: Adding a constant

Five friends took a 10 question multiple choice quiz in class. Their raw scores on the quiz are shown in the dotplot below along with summary statistics.
A dot plot has a horizontal axis labeled, Quiz scores, marked from 0 to 10, in increments of 1. The number of dots above each value is as follows: 6, 1; 7, 1; 8, 1; 9, 1; 10, 1. All values estimated.
x, with, \bar, on tops, start subscript, x, end subscriptstart text, M, end textstart text, I, Q, R, end textrange
Scores81, point, 41834
The teacher told everyone that she would add 1 to every student's score as extra credit. Their new scores are shown below.
A dot plot divided into 2 parts labeled Raw score and New score, has a horizontal axis labeled, Quiz scores, marked from 0 to 11, in increments of 1. The portion of the dot plot dedicated to the Raw score has dots above values as follows: 6, 1; 7, 1; 8, 1; 9, 1; 10, 1. The portion of the dot plot dedicated to the New score has dots above values as follows: 7, 1; 8, 1; 9, 1; 10, 1; 11, 1. All values estimated.
Problem A (part 1)
Find the summary statistics for the new scores.
Hint: You don't need to calculate these using their formulas. Look at the dotplots to visually assess the center and spread of each distribution.
x, with, \bar, on tops, start subscript, x, end subscriptstart text, M, end textstart text, I, Q, R, end textrange
Raw scores81, point, 41834
New scores
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

problem b (part 1)
Describe the effects adding the constant had on the measures of center and spread.
Choose 1 answer:

Part 2: Multiplying a constant

The teacher always scores her quizzes out of 100 points. For this 10-question quiz, she multiplies the new scores by 10 to get the students' final grades which are shown in the dotplot below.
x, with, \bar, on tops, start subscript, x, end subscriptstart text, M, end textstart text, I, Q, R, end textrange
Scores81, point, 41834
New scores91, point, 41934
Final gradesquestion markquestion markquestion markquestion markquestion mark
problem a (part 2)
Find the mean and median of the final grades.
Hint: You don't need to calculate these using their formulas. Think about how the center has or hasn't changed.
mean =
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
median =
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

problem b (part 2)
Find the range of the final grades.
range =
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

problem c (part 2)
Considering what happened to the range, guess what the IQR and standard deviation are for the final grades.
Hint: You don't need to calculate these using their formulas. Think about how the spread has or hasn't changed.
start text, I, Q, R, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
s, start subscript, x, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

problem d (part 2)
Describe the effects multiplying these data by a constant had on the measures of center and spread.
Choose 1 answer:

Part 3: Putting it all together

Let's look at a temperature conversion example. Suppose a set of temperature measurements has a mean of 104, degrees, start text, F, end text and a standard deviation of 9, degrees, start text, F, end text, and we convert all of the temperatures to degrees celsius.
Here's the conversion formula: degrees, start text, C, end text, equals, left parenthesis, degrees, start text, F, end text, minus, 32, right parenthesis, times, start fraction, 5, divided by, 9, end fraction
problem A (part 3)
What is the mean of the temperatures in degrees celsius?
mean =
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees, start text, C, end text

problem B (part 3)
What is the standard deviation of the temperatures in degrees celsius?
standard deviation =
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees, start text, C, end text

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  • starky seed style avatar for user brandosantanacasaslopez
    Posted 6 years ago. Direct link to brandosantanacasaslopez's post “Why subtracting does not ...”
    Why subtracting does not impact the standard deviation?

    Could anyone break it down?
    (12 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Vincent
      Posted 6 years ago. Direct link to Vincent's post “Shifting all of the data ...”
      Great Answer
      Good Answer
      Shifting all of the data points left or right on a number line doesn't impact the standard deviation of a data set. This is because standard deviation is a measure of how spread out data points are. Because adding and subtracting don't change the spread of the data, the standard deviation doesn't change. This also holds true for the range and IQR.
      (43 votes)
  • blobby green style avatar for user Jenifer De Figueiredo
    Posted 6 years ago. Direct link to Jenifer De Figueiredo's post “I'm confused about why th...”
    I'm confused about why this is the answer. What is the math/logic behind leaving one part of the conversion problem out of the scenario when dealing with SD? I see that it works, and get that we're measuring distances from the mean in degrees, not actual degrees, so subtracting 32 wouldn't make much sense and it would make the answer negative, which doesn't work for SD. But why does the multiplying by 5/9 part work? I guess what I'm confused about is what is the 5/9ths part to the temperature conversion formula?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Anne Mylott
      Posted 6 years ago. Direct link to Anne Mylott's post “One way to think about th...”
      Good Answer
      One way to think about this is to compare to distance measurements. If you look at a ruler, you will see that inches are wider than centimeters. If you look at a thermometer, you will see that degrees Celsius are bigger (sort of "wider") than degrees Fahrenheit.
      A ruler starts counting both inches and centimeters in the same place, so the conversion formula only involves multiplication (0 inches = 0 cm; 1 inch = 2.5 cm, 2 inches = 5 cm; etc). Nothing would be added or subtracted from the mean or the SD.

      But a thermometer starts counting in different places: 0 C = 32 F. That's why we add or subtract 32 when converting temperatures (how much warmer is it than the freezing point of water). Mean is a temperature (how warm it is on an average day), so we have to add or subtract 32 when converting between C and F.
      Standard deviation does not measure how hot it is, but rather how different a set of days are from the average day. SD is a measure of spread. (In different places, the same "width" of spread could be around a mean temperature 0 C, 20 C, or 1000 C.) Because of that, the only thing that matters is that a degree Celsius is 9/5 "wider" than a degree Fahrenheit, aka, a degree Fahrenheit is 5/9 "narrower" than a degree Celsius. That's why we only multiply the units, we don't add or subtract.

      Another comparison to reinforce the point: The Kelvin and Celsius scales use the same size ("width") units, but zero Kelvin is "absolute zero", the coldest possible temperature. Zero Celsius is the freezing point of water. Same units, just shifted by hundreds of degrees. What happens to the mean and standard deviation when you convert between Kelvin and Celsius? [For an explanation of absolute zero, take a physics course; here just trust me it's waaaay colder than 0 C. Right now, focus on the math, which is all you need to answer my questions.]
      [pause to think, then scroll down for answers and a new question]




      Answers: No multiplication involved. The mean shifts but the SD stays the same. Why?
      (24 votes)
  • blobby green style avatar for user Merrill Van Gerpen
    Posted 4 years ago. Direct link to Merrill Van Gerpen's post “How do you arrive at 3 fo...”
    How do you arrive at 3 for the IQR? Quatile one equals 7 and quartile three equals 9, so wouldn't that make the IQR equal to 2?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user N N
    Posted a year ago. Direct link to N N's post “I don't understand how to...”
    I don't understand how to convert a standard deviation of 9F to degrees celsius.

    C = (F-32) * 5/9

    Why we can replace (F-32) with a standard deviation of 9F?
    Why we don't calculate using the formula? : C = (9F-32)*5/9
    (3 votes)
    Default Khan Academy avatar avatar for user
  • leafers sapling style avatar for user green_ninja
    Posted 2 years ago. Direct link to green_ninja's post “Hi! Is the range affecte...”
    Hi!

    Is the range affected when a set is shifted?

    Also, for the last problem concerning standard deviation, why don't we subtract 32? Doesn't that result in an incorrect conversion?

    Thanks!
    (2 votes)
    Default Khan Academy avatar avatar for user
  • duskpin sapling style avatar for user CassandraBillings
    Posted 5 years ago. Direct link to CassandraBillings's post “So when given a problem w...”
    So when given a problem where you must find the standard deviation or mean, how would you find that?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • purple pi purple style avatar for user Luis Fernando Hoyos Cogollo
    Posted 5 years ago. Direct link to Luis Fernando Hoyos Cogollo's post “Part 2: Multiplying a con...”
    Part 2: Multiplying a constant

    If it's a 10-question quiz, and every grade is multiplied by 10, no result should be more than 100.
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user hayate.takeda
    Posted 3 years ago. Direct link to hayate.takeda's post “I understand that multipl...”
    I understand that multiplication changes the SD and all, but I don't understand why addition/subtraction in the case of temperatures doesn't matter. 9 degrees Fahrenheit IS -12.78 degrees Celsius, so why is it 5 degrees Celsius?
    (1 vote)
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  • winston baby style avatar for user yes
    Posted 4 months ago. Direct link to yes's post “For the standard deviatio...”
    For the standard deviation, the original value for the temperature conversion was C = (F - 32)* 5/9. Are the mean and standard deviation different when calculating the temperature conversion?
    (1 vote)
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  • blobby green style avatar for user Anand Jha
    Posted 4 months ago. Direct link to Anand Jha's post “when we are changing the ...”
    when we are changing the scale itself then why the subtraction is not considered in formula?
    (1 vote)
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    • leaf green style avatar for user Yahya
      Posted 4 months ago. Direct link to Yahya's post “You would get a negative ...”
      You would get a negative number if you do. The spread is for measuring how far apart the numbers are and it makes sense if you look at data with all the values converted. The temperatures in F are more spread apart like how 100 F is 37.8 C and 60 F is 15.6 C. The spread is bigger in Fahrenheit. The difference is 40 in F but if you multiply by 5/9 you get 22.2 and 22.2 is also the difference between 37.8 and 15.6 in Celsius. The subtraction part when you only convert the temperature is because they also start from a different point, 32 F for freezing but with Celsius it's 0. For measuring spread it's not measuring from a certain point but just the differences between the values.
      (1 vote)