Chapter #8 - 10/21/24 Flashcards
bell shaped curves and other shapes
Explain what is meant by a frequency curve for the possible values of measurement variable :
A frequency curve is a smooth line that shows the overall pattern of data for a measurement variable, like height, weight, or test scores. It tells us how often each possible value occurs across the range of data.
Describe the characteristics of a bell-shaped frequency curve :
A bell-shaped frequency curve, also called a normal distribution, has these key characteristics:
- Symmetry: It’s perfectly symmetrical around the center.
- Mean = Median = Mode: All three are located at the center peak.
- Tails: The curve gradually tapers off on both sides, extending towards infinity without touching the axis.
- 68-95-99.7 Rule: About 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Explain how the Empirical Rule is used to provide information about a normal distribution with specific mean and standard deviation :
The Empirical Rule helps us understand how data is distributed in a normal distribution:
- 68% of data falls within 1 standard deviation of the mean.
(This means that about 68% of values are between the mean minus one standard deviation and the mean plus one standard deviation.)
- 95% of data falls within 2 standard deviations of the mean. (So, about 95% of values lie between the mean minus two standard deviations and the mean plus two standard deviations.)
- 99.7% of data falls within 3 standard deviations of the mean. (Nearly all values (99.7%) are between the mean minus three standard deviations and the mean plus three standard deviations.)
Suppose you are taking a class with 200 students and you learn that your score on the last exam was at 70th percentile. What does that mean?
how many students in class were below you
what is a “standardized score”?
the number of standard deviations an individual falls above or below thte mean for the whole group
what is a “bell shaped curve” ? what does it look like ?
what are frequency curves ?
Smoothed-out histogram by connecting tops of rectangles with smooth curve
TRUE OR FALSE
all frequency curves are bell-shaped?
FALSE
not all frequency curves are bell-shaped
describe a frequency curve that follow a right skewed distribution :
values to the left more clumped together; values to the right more spread out
describe a frequency curve that follow a left skewed distribution :
values to the right more clumped together; values to the left more spread out
describe mean and median in regards to a right skewed frequency curve :
mean > median
describe mean and median in regards to a right skewed frequency curve :
mean < median
FILL IN THE BLANK
the long right tail pulls the mean to the _______
right
what does the total area frequency curve =
1 for 100%
proportion of population of measurements falling in a certain range =
area under curve over the range
when does the mean = the median ?
in symmetric distributions
economy/money usually equals what ?
skewed to the right usually
height usually equals what ?
symmetric
many populations of measurements follow approximately a normal curve :
- physical measurements within a homogeneous population - heights of male adults
- many psychological attributes (IQ)
- standard academic tests given to a large group (SAT scores)
all normal curves have the same shape :
symmetric, unimodal and bell-shaped
FILL IN THE BLANK
normal curves are completely describe by giving its ______ and _____________
mean μ and standard deviation σ
where is the mean located in normal curves ?
at the center of the curve and is the same as the median
in a normal curve, what does changing the mean do to the cuvre ?
moves the curve along the horizontal line (increasing the standard deviation makes the area under the curve to be less concentrated around the mean)
what does your percentile = ?
the percentage of the population that falls below you
what does finding percentiles for nomal curves require :
- your own value
the mean for the population of value (μ) - the standard deviation for the population (σ)
TRUE OR FALSE
the standard normal (bell-shape) curve can be used to find percentiles
TRUE
what is another name for standardized score ?
z-score
what are IQ scores with a noraml distributions mean and standard deviation ?
IQ scores have a normal distribution with a mean of 100 and a stadard deviation of 15
what is the formula for standardized score ?
χ - μ / σ
what does “χ” represent ?
observed value
what does “μ” stand for ?
mean
what does “σ” stand for ?
standard deviation
suppose you IQ score was 115, what is you standardized score ? (mean of 100 and standard deviation of 15)
(115 - 100) / 15 = +1
if your IQ score is +1, what does this mean ?
your IQ is 1 standard deviation above the mean
what is a normal curve with a mean = 0 and a standard deviation = 1 called ?
standard normal curve
what is a z table ?
a chart that shows the probability, or area, under a standard normal curve (bell curve) for different z-scores. A z-score represents how many standard deviations a data point is from the mean. By looking up a z-score in the z-table, you can find the percentage of data that falls below that score in a normal distribution.
how do you read a z table ?
So, for a z-score of 1.28, you’d look up 1.2 on the left and 0.08 on the top, which will give you a value close to 0.8997. This means 89.97% of the data lies to the left of a z-score of 1.28.
how to find a percentile from an observed value ?
1) find the standardized score = (bserved value - mean) / s.d., where s.d. = standard deviation (dont forget to keep the plus or minus sign)
2) look up the percentile in table
1) Suppose your IQ score was 115
2) Standardized score = (115 – 100)/15 = +1
3) Your IQ is 1 standard deviation above the mean.
4) From Table 8.1 you would be at the 84th percentile.
what would you IQ be higher than ?
that of 84% of the population
what is the empirical rule ? (for any normal curve)
- 68% of the values fall within 1 standard deviation of the mean in either direction
- 95% of the values fall within 2 standard deviations of the mean in either direction
- 99.7% of the values fall within 3 standard deviations of the mean in either direction
when would a measurement be considered an extreme outlier ?
if it fell more than 3 standard deviations above or below the mean
what is the empirical rule ?
The empirical rule is a guideline for data that follows a normal distribution (bell-shaped curve). It says:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
