1. Check it out!
1.2.3: Assessing Normality

2. Domenic decided to research the word lengths in his favorite novel for his statistics class. He used a sampling technique to select 40 words from the book. Then, he created a table of the sample words and the number of letters in each word, and used it to create a histogram. Use Domenic’s table and histogram, shown on the next two slides, to solve the problems that follow.
Common Core Georgia Performance Standard: MCC9–12.S.ID.4★
1.2.3: Assessing Normality

3. Sample word Number of letters a he it no of to few lay one Jon she1
the 3
seemed 6 he 2
was
vermin
outside 7 it
call 4
shaking no
door
sliding of
hope
tonight to
like
attempts 8
roof
distinct
room
greatest
few
said
forcefully 10
lay
with
extremities 11
one
clips 5
anticipation 12
Jon
skill
she
Arthur
1.2.3: Assessing Normality

4. Word Lengths Histogram
1.2.3: Assessing Normality

5. What percent of words in the sample have more than 10 letters?
Estimate the number of 3-letter words used in the book. Why do you think there are so many 3-letter words?
Find the mean and median word length for the sample.
Find the standard deviation of the word lengths in the sample.
Are the word lengths in this book symmetric? Justify your answer.
1.2.3: Assessing Normality

6. What percent of words in the sample have more than 10 letters?
There are 40 words in the sample, and 2 of them have more than 10 letters.
5% of the words in the sample have more than letters.
1.2.3: Assessing Normality

7. Estimate the number of 3-letter words used in the book
Estimate the number of 3-letter words used in the book. Why do you think there are so many 3-letter words?
Assuming the sample words are representative of the words in the book, we can use the percent of 3-letter words in the sample to estimate the percent of 3-letter words in the population (all the words in the book).
1.2.3: Assessing Normality

8. 20% of the words in the book are estimated to be 3-letter words
20% of the words in the book are estimated to be 3-letter words. There are several possible reasons for why there are so many 3-letter words. For example, the sample includes words such as the and was that are commonly used in the English language. There are also several 3-letter pronouns and prepositions.
1.2.3: Assessing Normality

9. Find the mean and median word length for the sample.
The mean is the sum of the data points divided by the number of data points.
The mean of this data set is 4.65 letters per word.
The median is the middle number, or the average of the two numbers closest to the middle, when the numbers are arranged in order from lowest to highest (or highest to lowest).
1.2.3: Assessing Normality

10. Since there are 40 words, the two numbers closest to the middle are the twentieth and twenty-first words in the list. Each of these words (hope and like) has 4 letters, so the median is 4 letters per word.
1.2.3: Assessing Normality

11. Find the standard deviation of the word lengths in the sample.
The formula for calculating standard deviation is
Use a calculator to find the standard deviation of the word lengths in the sample.
σ ≈ 2.7 words
1.2.3: Assessing Normality

12. Are word lengths in this book symmetric? Justify your answer.
No. The word lengths are not symmetric because the percentage of data that lies above the mean is different from the percentage of data that lies below the mean. In this sample, of the word lengths in the sample are below the mean, while only 37.5% are above the mean. We expect something closer to a 50–50 split for symmetric distributions.
Connection to the Lesson
The warm-up data set reviews proportion, mean, median, and standard deviation, which are used in the lesson.
Students will be analyzing histograms to determine if a data set is normally distributed.
1.2.3: Assessing Normality