Download median

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 =
87.0769308
3.6366666
• Median = 88
• Median = 3.65
• Mode = 97 (x3)
• Mode = None
• Stand. Deviation =
• Stand. Deviation =
10.08015215
0.2058370445
Normal Distribution
Mean = Median
“Bell Curve”
Mean + Median
Skewed data is described based on location of tail
Positively Skewed (Right)
Negatively Skewed (Left)
“Mean is to Right of Median”
Median Mean
“Mean is to Left of Median”
Mean Median
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
not adequately represent the population.
– 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.
BIASED
• 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.
UNBIASED
• Ex 3: Putting the names of all the seniors in a hat, then drawing
names from the hat to select a sample of seniors.
UNBIASED
• Ex 4: Determining the shopping preferences of the students at
your school by asking people at the mall.
BIASED
• Ex 5: Finding the average height of the students in your school by
using the members of the football team.
BIASED
• 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”.
– the margin of error
0.38(1  0.38)
ME  2
 .025 (ME) is 2.5%
1500
• That means that there is a 95% chance that the people in the
population that would answer “Yes” between (38-2.5) =
35.5% and (38 + 2.5)= 40.5% of the time.
• (38 % with a margin of error of 2.5%)
Exp 2:. p = 16%, n = 400 Exp 3: 934 out of the 2150
0.16(1  0.16)
ME  2
400
 .0367
students said they read the
newspaper every day
934
934
(
1

2150
2150 )
ME  2
 .0214
2150