- If a population has the following characteristics
- The mean, median and mode are equal
- the histogram is symmetric about the mean
- The it is likely that it is described by the normal distribution or
- The Gaussian distribution
- The bell curve
- Many populations can be modeled by the normal distribution.
- In a normal distribution
- 68% of the data fall within 1 standard deviation of the mean (both sides combined, 34% from mean to +1 sd and 34% from mean to -1 sd.
- 95% of the data fall within 2 standard deviations.
- 99.7% fall within three standard deviations.
- Example: Do exercises 12-22 on page 680
12 Between $16,000 and $18,000.
This is the data that is within
two standard deviations of the
mean, so 95% of the data is in this range.
14 Between 17,000 and 18,000
This is 1/2 of the total between two
standard deviations of the mean
so 95/2% or 47.5% of the data is in this range.
20 More than 18,000
50% of the data is less than 17,000
47.5% of the data is between 17,000 and 18,000
so 97.5% of the data is less than 18,000
so 100%-97.5% is more than 18,000,
Or 2.5%
- A z-score is computed by
data item - mean
z-score = ------------------
standard deviation
- If the z-score is
- negative, the data item is below the mean
- zero, the data item is at the mean
- positive, the data item is above the mean
- Example:
s = 8, x = 60.
#34: Data = 76
z-score = 76-60/8 = 16/8 = 2
#38: Data = 72
z-score = 72-60/8 = 12/8 = 1.5
#44: Data = 44
z-score = 44-60/8 = -16/8 = -2
- Example:
s = 50, x = 400.
solving for data we get:
data = z*s-x
# 60: z = 3
data = 3*50+400 = 550
# 66: z = -1.5
data = -1.5*50+400 = 325
- If we have a z-score, we can use table 12.10
to determine the percentage of the data the is below that z-score.
- Example:
# 68: z= .8
Look this up in the table: 78.18
so 78.81% of the data is below this score
and 100-78.81 = 21.19% of the data is above.
- Example:
#82, between z=-2.2 and z=-0.3
Look up both numbers: 1.39% and 38.21%
38.21 - 1.39 = 36.82%