calculating area under the density curve
The total area under any valid density curve is always equal to this number.
What is 1 (or 100%)?
his formula calculates a standard score by subtracting the mean from an individual observation and dividing by this population or sample value.
What is the standard deviation? ($z = \frac{x - \mu}{\sigma}$)
In a distribution that is strongly skewed to the left, this primary measure of center is typically lower than the median.
What is the mean?
The two parameters that completely specify any Normal distribution curve.
What are the mean ($\mu$) and standard deviation ($\sigma$)?
When a constant $c$ is added to every value in a data set, measures of center (mean, median) are affected in this way.
What is they increase by $c$?
If a continuous uniform density curve spans horizontally from $x = 0 to $x = 5, the height (ordinate) of the rectangle must be equal to this.
What is 0.2 (or 1/5)?
A value that separates the bottom 80% of a distribution from the top 20% is known by this term.
What is the 80th percentile?
A graphical distribution of quantitative data that features two distinct, prominent peaks is described by this statistical term.
What is bimodal?
According to the Empirical Rule (68-95-99.7 rule), this percentage of a Normal distribution falls between $\mu - 2\sigma$ and $\mu + 2\sigma$.
What is 95%?
If every value in a data set is multiplied by a constant $b$ (where $b > 0$), measures of center (mean, median) and measures of position (percentiles) are affected in this specific way.
What is they are also multiplied by $b$?
To find the proportion of observations less than a specific value on a continuous density curve, you calculate this geometric or integral concept. Zip codes of students living in St. Johns County.
What is the area under the curve to the left of that value?
If a test score has a z-score of -1.5, it means the score is this many standard deviations relative to the mean.
What is 1.5 standard deviations below the mean?
Suppose a data set has a median of 50 and a mean of 68, and the data values are heavily crowded on the lower end with a long tail stretching to the right. State the direction of the skewness and explain why the mean is pulled away from the median.
What is skewed to the right (positively skewed)? (The high extreme values in the right tail pull the arithmetic mean upward, whereas the median is more resistant).
A Normal distribution with a mean of 0 and a standard deviation of 1 has this special designation.
What is the Standard Normal Distribution ($N(0,1)$)?
If every value in a data set is multiplied by 3 and then 5 is added to it, the variance is multiplied by this factor.
What is 9? (Since variance is squared units, multiplying by 3 scales the variance by $3^2 = 9$; adding a constant has zero effect on variance).
For a continuous uniform distribution defined between x = 10 and x = 30, what is the proportion (area) of observations greater than x = 25?
What is 0.25 (or 25%)? (The interval from 25 to 30 has a width of 5, and the uniform density height is 1/(10-30=0.05, so Area - 5 x 0.05 = 0.25).
Given a distribution with a mean of 70 and a standard deviation of 8, this is the raw data score corresponding to a z-score of 1.25.
What is 80? (x = mu + zxsigma = 70 + 1.25(8) = 70 + 10 = 80)
If you add 10 to every single value in a data set, the shape and skewness of the distribution undergo this change.
What is no change? (Adding a constant shifts the center horizontally, but does not alter the shape or spread).
In a factory producing rods, lengths are normally distributed with a mean of 100 cm and a standard deviation of 2 cm. Using the Empirical Rule, what percentage of rods will have a length less than 96 cm?
What is $2.5\%$? (96 cm is two standard deviations below the mean, $\mu - 2\sigma$. By the rule, 95% fall between $\pm 2\sigma$, leaving 5% in both tails combined, so $5\% / 2 = 2.5\%$ in the lower tail).
If every value in a quantitative data set is divided by a positive constant $c$, how are the measures of center (mean, median) and measures of spread (standard deviation, IQR) affected?
What is they are all divided by $c$? (Division by a positive constant scales both the center and the spread down by that exact factor).
If the area under a density curve between x = 1 and x = 3 is 0.45, and the area to the left of x = 1 is 0.20, this is the cumulative proportion to the left of x = 3. Explain why "shirt size (Small, Medium, Large)" is considered categorical rather than quantitative, despite having a logical order.
What is 0.65 (0.20 + 0.45)?
If two students take completely different tests and both score in the 90th percentile of their respective classes, this explains why their actual raw scores or z-scores might differ.
What is that percentiles represent relative standing (rank) within each specific reference group, not absolute score values?
Describe the exact relationship between the mean, median, and mode in a perfectly symmetric, unimodal distribution.
What is they are all equal (or located at the exact same point)?
Scores on a statewide exam are normally distributed with an unknown mean and a standard deviation of 12. If a student scores an 82 and their corresponding $z$-score is $+1.5$, what is the mean ($\mu$) of the exam?
What is $64$? (Using $z = \frac{x - \mu}{\sigma} \rightarrow 1.5 = \frac{82 - \mu}{12} \rightarrow 18 = 82 - \mu \rightarrow \mu = 64$).
A data set has a mean of 60 and a standard deviation of 10. If every data point is multiplied by $2.5$ and then $15$ is subtracted, what are the resulting new mean and new standard deviation?
What is a new mean of $135$ and a new standard deviation of $25$? (New mean = $2.5(60) - 15 = 150 - 15 = 135$; New standard deviation = $2.5(10) = 25$, since subtracting a constant does not change spread).