Density Curve
What is the purpose of the density curve?
The density curve describes the overall pattern of a distribution and is intended to reflect the idealized shape of the population distribution.
What happen to the curve when the std ( sigma) gets bigger?
When σ is larger, the curve spreads out further and the area under the Normal curve is less concentrated about the mean.
How many percents of the observations fall within one standard deviation of the mean: (µ − σ, µ + σ).
68%
What percent of high school seniors meet this SAT requirement of a combined score of 980 or better?Given: N(µ = 1059, σ = 210).
0.648
What is the difference between a density curve and a histogram?
A histogram is a plot of data obtained from a sample. A density curve is a reflection of data distribution under a curve
1=100%
What does the STD σ control?
STD controls the variability of the variability or shape of a Normal curve
What is a z- score( standard score)?
It is the result of the process called standardizing? You take the observation - the mean and altogether divided by the std
In a course with 500 students, the midterm scores are N(µ = 70, σ = 10), want to find the proportion of students receive a score below 84.7. What is the result ?
92.92%
The distribution of heights of women aged 18 to 24 is approximately Normal with mean 63.7 inches and a standard deviatoin of 2.5 inches.
Q2. What percentage of young women have heights between 58.7 and 66.2 inches? i.e. P(58.7
81.5%
Two important properties of a density curve?
1) is always on or above the horizontal axis.
2) has an area of exactly 1=100% underneath it.
Where is the mean µ is located in the curve? Where is curve move when you change the mean without changing the standard deviation?
The mean µ is located at the center of the symmetric curve and is the same as the median.
Changing µ without changing σ moves the Normal curve along the horizontal axis without changing its variability.
Where is the positive/negative z score located at?
A positive z-score indicates the observation is above mean.
A negative z-score indicates the observation is below the mean.
The distribution of heights of women aged 18 to 24 is approximately Normal with mean 63.7 inches and a standard deviatoin of 2.5 inches.
Q1. What percentage of young women have heights above 66.2 inches? i.e. P(women height > 66.2)?
16%
Describe the special Normal distribution?
The standard Normal distribution is the Normal distribution with mean µ = 0 and standard deviation σ = 1, denoted as N(0, 1).
where is mean and median in a density curve?
- The median of a density curve is the equal-areas point, which divides the area under the curve in half.
- The mean of a density curve is the balance point, or center of gravity, at which that curve would balance if it were made of solid material.
-The median and mean are the same for a symmetrical density curve. They both lie at the center of the curve.
What is the shape of a normal distribution?
symmetric, single-peaked, and bell-shaped
What is c-th percentile of a distribution means?
The c-th percentile of a distribution is a value such that c percent of the observations lie below it. That is, P(observations ≤ c-th percentile) = c ∈ [0, 1]. The median of any distribution is the 50% percentile.
The heights of all women aged 20 to 29 in the US are approximately Normal with µ = 64.1 and σ = 3.7 inches. The standardized height is
-1.11
What are the two determinant values of a normal curve?
A specific Normal curve is completely determined by its mean and standard deviation
Where is the mean of a skewed curve?
The mean of a skewed surve is pulled away from the median in the direction of the long tail.
What is standardizing?
Mapping observations from N(µ, σ) units to N(0, 1) units.
How many percent of the observations fall within three standard deviations of the mean: (µ − 3σ, µ + 3σ).
99.7%
How high must a student score in order to be in the top 10% (90% percentile) of the distribution? i.e. find x0 so P(scores ≥ x0) = 10%
Given: SAT reading scores for a recent year are distributed according to an N(µ = 531, σ = 104) distribution.
x0 = 664.12
If a distribution is skewed to the left? Where is the mean and the median?
The mean is less than the median.