The document introduces the concept of the normal distribution and how it relates to many real-world datasets that follow a bell curve shape. It describes that a normal distribution is defined by its mean and standard deviation. It then validates the empirical rule for normal distributions, which states that approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. The document tests this by analyzing data on baby weights and calculating the percentages within each standard deviation range.

1. Introduction to Normal
Distribution

2. Consider the following data pertaining to hospital weights (in pounds) of
all the 36 babies that were born in the maternity ward of a certain
hospital.
The data have an average of 6.11 pounds and a standard deviation of
1.61 pounds.
4.94 4.69 5.16 7.29 7.19 9.47 6.61 5.84 6.83
3.45 2.93 6.38 4.38 6.76 9.01 8.47 6.8 6.4
8.6 3.99 7.68 2.24 5.32 6.24 6.19 5.63 5.37
5.26 7.35 6.11 7.34 5.87 6.56 6.18 7.35 4.21

3. Histogram for Babies’ Weights Dataset
Observe that the histogram is approximately bell-shaped

4. Normal Distribution
• Many continuous random variables, such as IQ scores, heights of
people, or weights of M&Ms, have histograms that have bell-shaped
distributions
• The most important distribution in statistics is a normal distribution,
which has a "bell-shaped" curve
• The normal distribution is a continuous distribution
• However, the left and right tails of the normal distribution extend
indefinitely but come infinitely close to the x-axis

5. Illustration of Normal Curve
• graph of the normal distribution depends on two factors: the mean 𝜇
and the standard deviation σ
• the mean determines the location of the center of the bell shaped
curve
• Thus, a change in the value of the mean shifts the graph of the normal
curve to the right or to the left.

6. Mean, Median, Mode of a Distribution
• Mean – represents the balancing point of the graph of the distribution
• Median - represents the point where 50% of the area under the
distribution is to the left and 50% of the area under the distribution is
to the right
• Mode - represents the “high point” of the probability density function
(i.e. the graph of the distribution)

7. Types of Normal Distribution
Symmetric: Mean=Median=Mode
Skewed to the right (Positively skewed): Mean>Median>Mode
Skewed to the left (Negatively skewed): Mean<Median<Mode

8. Standard Deviation (𝜎)
• standard deviation determines the shape of the graphs (particularly, the height
and width of the curve)
• When the standard deviation is large,
the normal curve is short and wide
• a small value for the standard deviation
yields a skinnier and taller graph

9. Property of Normal Distribution
The total area under the normal curve is equal to 1.

10. Empirical Rule
• It is actually a theoretical result based on an analysis of the
normal distribution.
• Also called 68-95-99.7 rule
• Every normal curve (regardless of its mean or standard
deviation) conforms to the following “empirical rule”

11. Empirical Rule
• About 68% of the area under the curve falls within 1 standard deviation of the
mean.
• About 95% of the area under the curve falls within 2 standard deviations of the
mean.
• Nearly the entire distribution (About 99.7% of the area under the curve) falls within
3 standard deviations of the mean.

12. Empirical Rule

13. Empirical Rule

14. Consider the following data pertaining to hospital weights (in pounds) of
all the 36 babies that were born in the maternity ward of a certain
hospital.
The data have an average of 6.11 pounds and a standard deviation of
1.61 pounds.
4.94 4.69 5.16 7.29 7.19 9.47 6.61 5.84 6.83
3.45 2.93 6.38 4.38 6.76 9.01 8.47 6.8 6.4
8.6 3.99 7.68 2.24 5.32 6.24 6.19 5.63 5.37
5.26 7.35 6.11 7.34 5.87 6.56 6.18 7.35 4.21

15. Validating the Empirical Rule
Determine the frequency and relative frequency (percentage) of babies’
weights that are:
• Within one standard deviation from the mean
• Within two standard deviations from the mean
• Within three standard deviations from the mean
• Below the mean
• Above the mean

16. Validating the Empirical Rule
• One standard deviation from the mean
• Two standard deviations from the mean
• Three standard deviations from the mean
• Below the mean
• Above the mean
Answers:
• 26 out of 36; or about 72% of the data within one 𝜎 from 𝜇
• 34 out of 36; or about 94% of the data within two 𝜎 from 𝜇
• 36 out of 36; 100% of the data within three 𝜎 from 𝜇
• 15 out of 36; or about 42% of the data is below the mean
• 20 out of 36; or about 56% of the data is above the mean

17. PRACTICE:
The following data pertains to the points scored of the high school
basketball team in 28 games. The standard deviation is 2.28
What frequency and relative frequency (percentage) of the data are:
• Within one standard deviation from the mean
• Within two standard deviations from the mean
• Within three standard deviations from the mean
• Below the mean
• Above the mean
66 75 41 57 54 82 67
42 60 37 49 87 101 78
60 66 48 43 42 61 64
67 37 51 63 68 77 13

18. Assignment
BRING A PRINTED STANDARD NORMAL TABLE