#### Transcript Mean

```UNIT 8:Statistical Measures
Statistics practice of analyzing a set of data
Measures of Central Tendency:
numbers that represent the middle of the data
Mean ( x ): Arithmetic average
(Total Data of Values over Number of values)
Median: Middle of the data listed in ascending order
(Count inward from Max and Min – if not an exact data
value, find midpoint of last two data values)
Mode:
Most commonly occurring number(s)
(MORE THAN ONE value if repeated same number of times
or NONE if no repeated values)
Measures of Variation: Variance, Standard Deviation
Dispersion: How spread / scattered a set of data is
Range: Difference between the highest and lowest data
value (MAX – MIN)
Inner Quartile Range (IQR):
The difference between Q3 and Q1 (Q3 – Q1)
Outlier: Any data item outside Q1 or Q3 a distance 1.5*IQR
*Standard Deviation (σ):
A measure of how much the data is spread out
**We don’t use variance**
Variance (σ2 ): Measures how much the data differs from the mean
5 Number Summary:
Min: Minimum Value (0 Percentile)
Q1: Quartile 1 (25th Percentile)
Med (Q2): Median (50th Percentile)
Q3: Quartile 3 (75th Percentile)
Max: Maximum or Q4 (100th Percentile)
Min
Med
Q1
Q2
Max
Q3
Q4
Calculator Commands:
One-Variable Statistics
• Input data: [STAT]  [EDIT]  L1
• [STAT] → [CALC] → [1: 1-Var STATS]
• σx → represents standard deviation
• x = represents mean
• n = number of entries
• med = median
• Etc…
Ex: Find the mean, median, mode, and
standard deviation of …
(1)Test Scores:
85, 76, 88, 91, 85, 58, 88,
91, 97, 91, 88, 97, 97
(2) GPAs: 3.42, 3.91,
3.33, 3.57, 3.45, 4.0, 3.65,
3.71, 3.35, 3.82, 3.67,
3.88, 3.76, 3.41, 3.62
• Mean =
• Mean =
• Median =
• Median =
• Mode =
• Mode =
• Stand. Deviation =
• Stand. Deviation =
Normal Distribution
Mean = Median
“Bell Curve”
Skewed data is described based on location of tail
Positively Skewed (Right)
Negatively Skewed (Left)
“Mean is to Right of Median”
“Mean is to Left of Median”
Normal Distribution:
68 – 95 – 99 Rule
x  Mean , Median
0.5%
2%
13.5%
34%
34%
13.5%
2% 0.5%
σ = Standard Deviation
1
_____of values are 1 st. deviation from the mean: ( x   , x   )
2
_____of values are 2 st. deviation from the mean:( x  2 , x  2 )
3
_____of values are 3 st. deviation from the mean:( x  3 , x  3 )
Normal Distribution Curve #1:
The number of problems missed on a quiz follows a
normal distribution with a mean of 15 and a standard
deviation of 4.
Draw a normal distribution for number of problems
missed.
3
7
11
15
19
(# wrong)
23
27
Related Questions:
What percent of students missed between 11 – 19
problems?
What percent of students missed more than 23
problems?
What percent of students less than 23 problems?
What percent of students missed fewer than 3
problems?
What percent of students missed less than 11 problems?
Normal Distribution Curve #2:
The final exam scores normal distribution with a mean
of 80 and a standard deviation of 6.
Draw a normal distribution for number of problems
missed.
(score)
Related Questions:
What percent of students scored between 80 - 92?
What percent of students scored between 68 -86?
What percent of students scored less than 98?
What percent of students scored higher than 74?
What percent of students scored less than 80?
Normal Distribution Curve #3:
The class mean of height is 60 inches. We know that
68% of the students are between 56 and 64 inches.
1) What is the standard deviation?
2) 50% of the students are shorter than …?
3) 16% of students are taller than …?
4) 2.5% of students are shorter than…?
5) 84% of students are taller than …?
Biased vs Unbiased
• When you are sampling a section of the population
(giving a poll), there are good and bad ways to do it.
• An unbiased sample is one in which a good section
of the population is represented.
– Entire population is well represented
• A biased sample is one in which the sample does
– A group or section of population is targeted or missing
The bigger the sample, the more accurate the results
(larger samples more closely reflect the population).
Determine if situation would produce (un)biased sample:
• Ex. 1: You want to determine how many people in a school are
going to college, so you ask every third person in an AP Calculus
class.
• Ex 2: You want to find out people’s favorite kind of food, so you
ask 100 people at the food court at the mall.
• Ex 3: Putting the names of all the seniors in a hat, then drawing
names from the hat to select a sample of seniors.
• Ex 4: Determining the shopping preferences of the students at
• Ex 5: Finding the average height of the students in your school by
using the members of the football team.
• The Margin of Sampling Error (ME) is a
numerical way to determine the difference between
how a sample and the population responds.
• If p represents the percentage of people with a
particular response from a sample of n people, then
95% of the time the population will respond within
one ME of the response p (p – ME or p + ME)
p(1  p)
ME  2
n
• Exp 1: 1500 people were asked a question and 38%
responded “yes”.
0.38(1  0.38) the margin of error (ME) is
ME  2
1500
• That means that there is a 95% chance that the people in the
population that would answer “Yes” between
Exp 2:. p = 16%, n = 400 Exp 3: 934 out of the 2150