Comparisons across normal distributions Z -Scores
Preview:
Citation preview
- Slide 1
- Comparisons across normal distributions Z -Scores
- Slide 2
- Overview Plan for the night Z-scores Definition Calculation Use
Graphing Data/Distributions Frequencies/Percentages
Charts/Graphs
- Slide 3
- Last time Last week we covered Measures of Central Tendency
Mean, Mode, Median Measures of Variability Range, IQR, SIQR,
Standard Deviation The most commonly used of the above are Mean
(SD) These two measures can be combined to further describe the
position of a score/datapoint
- Slide 4
- Is that a good score? Mean and SD are useful, but sometimes we
need to make comparisons between different measures Example (w/
same units of measure): SAT vs. ACT vs. GRE 10-yd dash time vs.
40-yd dash time Free-throw% vs. FG% vs. 3-Point% Example
(w/different unit of measure): ERA vs. WHIP VO 2max vs. Vertical
Jump BMI vs. %BodyFat vs. Waist Circumference
- Slide 5
- Minimal Statistics Mean SD m Z-scores Combine the mean w/ SD to
create a new unit of measurement (Standardizes Scores) Clearly
identifies a score as above or below the mean AND expresses a score
in units of SD Examples: z-score = 1.00 (1 SD above mean) z-score =
-2.00 (2 SD below mean) Describe the typical score, the spread of
scores, and the number of cases
- Slide 6
- Z-score = 1.0: GRAPHICALLY Z = 1 84% of scores smaller than
this Recall 50% of scores are below the mean + 34% of scores
between the mean and 1 SD above
- Slide 7
- Calculating z-scores Calculate Z for each of the following
situations: OR
- Slide 8
- Other features of z-scores 1) The Mean of a distribution of
z-scores = 0 Recall the mean is the balance point of a
distribution, where deviation scores sum to 0 A z-score of 0 is
equivalent to scoring the mean
- Slide 9
- Here is our normal distribution example from last week X = 70
SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = If a subject
scored 70, their z-score would be 0
- Slide 10
- Other features of z-scores 1) The Mean of a distribution of
z-scores = 0 Recall the mean is the balance point of a
distribution, where deviation scores sum to 0 A z-score of 0 is
equivalent to scoring the mean 2) The SD of a distribution of
z-scores = 1 Since SD is unit of measurement, when the mean is z=0
then the mean + 1 SD = a z-score of 1
- Slide 11
- Here is our normal distribution example from last week X = 70
SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = What is the
z-score of a subject that got: 80? 50? 100?
- Slide 12
- Other features of z-scores 1) The Mean of a distribution of
z-scores = 0 Recall the mean is the balance point of a
distribution, where deviation scores sum to 0 A z-score of 0 is
equivalent to scoring the mean 2) The SD of a distribution of
z-scores = 1 Since SD is unit of measurement, when the mean is z=0
then the mean + 1 SD = a z-score of 1 3) A z-score distribution is
same shape as raw score distribution Even though you are changing
the unit of measurement, this does not change the look of the
distribution when plotted
- Slide 13
- Here is our normal distribution example from last week X = 70
SD = 10 7060805090 40100 2.3% 34.1% 13.6% -3-20123Z = 34% of scores
still fall between 0 and 1 z-score
- Slide 14
- Z-score Comparison As stated, z-scores standardize different
distributions allowing you to make comparisons regardless of the
unit of measure Barts score SAT Exam 450 (mean 500, SD 100) Lisas
score ACT Exam 24 (mean 18, SD 6) Who scored higher? Bart: (450
500)/100 = - 0.5 Lisa: (24 18)/6 = 1
- Slide 15
- Z-scores & the normal curve For any z-score, we can
calculate the percentage of scores between it and the mean; all
scores below it & all above it Tons of online calculators:
http://www.measuringusability.com/normal_curve.php
- Slide 16
- What upper and lower limits include 95% of BMI scores? If one
boys BMI is 22 kg/m 2 and anothers WC is 70 cm, which of the two
has the highest adiposity? Example: Mean BMI and WC in elementary
school boys
- Slide 17
- Slide 18
- Nomenclature/Terminology Frequency: number of cases or subjects
or occurrences in a distribution Represented with f i.e. f = 12 for
a score of 25 12 occurrences of 25 in the sample
- Slide 19
- Nomenclature/Terminology Percentage: Number of cases or
subjects or occurrences expressed per 100 Represented with P or %
Ex. f=12 for a score of 25 when n=25 P = 12/25*100 = 48% (of scores
were 25)
- Slide 20
- Warning Should report the f when presenting percentages i.e.
80% of the elementary students came from a family with an income
< $25,000 different interpretation if n=5 compared to n=100
Reported in literature as f = 4 (80%) OR 80% (f = 4) OR 80% (n =
4)
- Slide 21
- Numerator Monster Pantagraph, 6/13/00 Pantagraph reported that
State Farm paid out over 1 Billion in dividends to customers in the
United States
- Slide 22
- Numerator Monster How much do you pay in car insurance every 6
months? Sohow much is State Farm keeping?
- Slide 23
- Frequency Distributions Graphically displaying the data should
ALWAYS come before any type of statistical analysis Measures of
central tendency and variability will give you a feeling for the
distribution of the data but its always easier to visually examine
it Check for normality (are data normally distributed?) Check for
outliers (are any subjects sticking out as odd?) Check of potential
associations (might two variables relate to each other?)
- Slide 24
- Frequency Distribution of Math Test Scores: SPSS Output 40
items on exam Most students >34 skewed (more scores at one end
of the scale)
- Slide 25
- Cumulative frequencies &, Cumulative percentages Cumulative
Percentage: how many subjects at and below a given score? i.e.,
33.3% of students scored a 32 or lower
- Slide 26
- Eyeball check of data: Intro to (brute force) graphing with
SPSS Stem and Leaf Plot: quick viewing of data distribution
Boxplot: visual representation of many of the descriptive
statistics discussed last week Bar Chart: frequency of all cases
Histogram: malleable bar chart Scatterplot: displays all cases
based on two values of interest (X & Y) Note: compare to our
previous discussion of distributions (normal, positively skewed,
etc)
- Slide 27
- Frequency Stem & Leaf 2.00 Extremes (=