Math Introduction 2014.ppt

 Statistics are used to describe data sets,
compare sets of data, and estimate how close
data is to describing all possible data
 Mean = Average value of all data
 Mode = Most common value in data set (occurs most
frequently)
 Median = Middle # in an ordered data set
 Arrange all #’s in order; pick middle most value
 Range = the span of a number set (Ex: lowest # through the
highest #; 10-30)
So how do you know if your hypothesis is
rejected? STATISTICS!!
X- Mean =
Number of samples
Sum of all Data

Variance (V)
 Means the variability of a population
 (how far a set of numbers is spread out ex. Variance=0 indicates all
values are identical)
 For each individual number in your data
set,
 Subtract the mean from each data set #
 square the results
 continue for each data point
 SUM ALL RESULTS
X= each individual data #
n= number of individuals measured (numbers of values in your data set)
Σ = Sum
X = mean

Standard Deviation (SD)
 A test of how close the data is to the mean
 Use SD to summarize the spread of values
around the mean.
 Low SD means all data close to mean
 Like throwing darts and getting almost all bullseyes
 Data is more consistent
 High SD means data spread widely from the mean
S = the square root
of variance
S = √variance

What is the formula for standard
deviation?
Note: since our data is a sample, not a
population, we will use n-1 and not just n for
our denominator.

What does standard deviation actually tell me?
The standard deviation tells you how tightly
your data is clustered around the mean.
When the bell curve is flattened (your data
is spread out), you have a large standard
deviation — your data is further away from
the mean.
When the Bell curve is very steep, your
data has a small standard deviation — your
data is tightly clustered around the mean.

0 2.816
2.023
-2.023
/2
/2
 A critical value is the value that a test statistic
must exceed in order for the the null
hypothesis to be rejected.
 A Null hypothesis says there is no significant
difference between the results obtained and the
results expected.
Critical Value and Null Hypothesis
Test statistic
Significance level (.05)
critical value
Null hypothesis: It, in essence, says that you propose that
nothing else-no factors-are creating the variation
in results except for random chance differences.

Why would we use the standard deviation to
analyze our lab result?
 In statistics and probability theory,
standard deviation (represented by
the symbol σ ) shows how much
variation or " dispersion " exists from
the average (mean, or expected value)
 A low standard deviation (less than 2) indicates
that the data points tend to be very close to the mean,
whereas high standard deviation (greater than 2)
indicates that the data points are spread out over a
large range of values.
http://www.mathsisfun.com/data/
standard-deviation.html

What are the basic steps find SD?
• Step 1-Find the mean of the data (in other
words, take an average of all your data
points)
• Step 2-Subtract the mean from each data
point
• Step 3- Square of the values you got in step 2
• Step 4-Subtract 1 from the number of data
points you used in step 1.
• Step 5- Divide your answer in step 3 by your
answer in step 4.
• Step 6 -Find the square root of the answer
you got in step 5.

~100% sure
3 X SE
95% sure 2 X SE
70% sure contains “real” mean 1 X SE
Standard Error (SE)
 A description of how close your data set got to “reality”
 You can never sample all possible outcomes; you can
only sample a small portion
 By taking into account the SD and sample size of
your data, SE can estimate how close you got to the
“real” mean
SE =
√Number of Samples
SD
Range of
Data Sample
Mean

T-Test
 A t-test is a statistical hypothesis test
to determine if a null hypothesis is
supported. It can be used to determine
if two sets of data are significantly
different from each other.

Calculating Confidence
1. Subtract 1 from the confidence level chosen (95%) and divide by 2.
Ex. 1-.95/2 =.025
2. Look up value on T-Table
Ex. Sample size is 10, so it is 10-1=9 degrees of freedom. The value on the T-
Table is 2.62
3. Take the Standard Error and multiply by the value found in Step 2.
Ex. .37x2.62 =.9694
4. Take the value from Step 3 and subtract it to the mean and then add it from
the mean. This will be your confidence limits.
Mean= 30 Confidence 30-.9694= 29.03 30+.9694=30.97
5. Do this for the other sample. If the confidence limits overlap then you would
have to accept the null hypothesis that there is no significant difference
between the two sample groups.

How to Graph Sample means with Standard
Error bars
 Find the sample
mean
 Find the standard
error of the mean
 Plot at 95%
confidence
( + 2SEM)
 Ex. SEM= 5 so you add 10 to the mean and
subtract 10 from the mean (multiple the SEM by
2)
 no overlap reject null hypothesis

Math Introduction 2014.ppt

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