Presentation on theme: "Data observation and Descriptive Statistics"— Presentation transcript:
1 Data observation and Descriptive Statistics We organize data so that it is easer to read and understand.
2 Organizing Data Frequency distribution Table that contains all the scores along with the frequency (or number of times) the score occurs.Relative frequency: proportion of the total observations included in each score.
3 Frequency distribution Amountf(frequency)rf(relative frequency)$0.0020.125$0.1310.0625$0.93$1.00$10.00$32.00$45.53$56.00$60.00$63.25$74.93$80.00$85.28$115.35$120.00n=161.00
4 Organizing data Class interval frequency distribution Scores are grouped into intervals and presented along with frequency of scores in each interval.Appears more organized, but does not show the exact scores within the interval.To calculate the range or width of the interval:(Highest score – lowest score) / # of intervalsEx: 120 – 0 / 5 = 24
5 Class interval frequency distribution f (frequency)rf ( relative frequency)$0-$246.375$25-$482.125$49-$733.1875$74-$98$99-$124n = 161.00
6 Graphs Bar graphs Data that are collected on a nominal scale. Qualitative variables or categorical variables.Each bar represents a separate (discrete) category, and therefore, do not touch.The bars on the x-axis can be placed in any order.
8 Graphs Histograms To illustrate quantitative variables Scores represent changes in quantity.Bars touch each other and represent a variable with increasing values.The values of the variable being measured have a specific order and cannot be changed.
9 HistogramNotice that the values on the x-axis have a specific order and cannot be rearranged.
10 Frequency polygon Line graph for quantitative variables Represents continuous data: (time, age, weight)
11 Frequency PolygonAGE22.0624.0525.0425.0726.0326.1127.0327.1129.0329.053437.153Make graph in classY – axis: frequencyX – axis: the scores, plot them with points and then connect the points.
12 Descriptive Statistics Numerical measures that describe:Central tendency of distributionWidth of distributionShape of distribution
13 Central tendency Describe the “middleness” of a data set Mean Median Mode
14 _ Mean Arithmetic average Used for interval and ratio data Formula for population mean ( µ pronounced “mu”)µ = ∑ X_____NFormulas for sample mean_X = ∑ X_____n
16 MeanNot a good indicator of central tendency if distribution has extreme scores (high or low).High scores pull the mean higherLow scores pull the mean lower
17 MedianMiddle score of a distribution once the scores are arranged in increasing or decreasing order.Used when the mean might not be a good indicator of central tendency.Used with ratio, interval and ordinal data.
21 Measures of Variability RangeFrom the lowest to the highest scoreVarianceAverage square deviation from the meanStandard deviationVariation from the sample meanSquare root of the variance
22 Measures of Variability Indicate the degree to which the scores are clustered or spread out in a distribution.Ex: Two distributions of teacher to student ratio.Which college has more variation?College ACollege B41612194122Sum = 57Mean = 19
23 Range The difference between the highest and lowest scores. Examples: Provides limited information about variation.Influenced by high and low scores.Does not inform about variations of scores not at the extremes.Examples:Range = X(highest) – X (lowest)College A: range = = 37College B: range = = 6
24 VarianceLimitations of range require a more precise way to measure variability.Deviation: The degree to which the scores in a distribution vary from the mean.Typical measure of variability: standard deviation (SD)VarianceThe first step in calculating standard deviation
25 Variance X = Number of therapy sessions each student attended. M = 4.2 “Deviation”Sum of deviations = 0
26 VarianceIn order to eliminate negative signs, we square the deviations.Sum the deviations = sum of squares or SS
27 Variance SD2 = Σ(X-M)2 N Take the average of the SS Ex: SS = 48.80 That is the average of the squared deviations from the meanSD2 = 9.76
28 ____ √ Standard Deviation Standard deviation Typical amount that the scores vary or deviate from the sample meanSD = Σ(X-M)2NThat is, the square root of the varianceSince we take the square root, this value is now more representative of the distribution of the scores.____√
29 Standard Deviation X = 1, 2, 4, 4, 10 M = 4.2 SD = (standard deviation)SD2 = 9.76 (variance)Always ask yourself: do these data (mean and SD) make sense based on the raw scores?
30 Population Standard Deviation The average amount that the scores in a distribution vary from the mean.Population standard deviation:(σ pronounced “sigma”)√____σ = ∑( X - µ ) ²_________N
32 Sample Standard Deviation Sample is a subset of the population.Use sample SD to estimate population SD.Because samples are smaller than populations, there may be less variability in a sample.To correct for this, we divide the sample by N – 1Increases the standard deviation of the sample.Provides a better estimate of population standard deviation.When we run experiments, we want to make sure the our results can generalize to the population at large. This also goes for the statistical procedures that we perform on the data of our sample.Differences in formulas from pop. to sample SD:Sigma is now “s”Mu is now “ X bar”and divide by N – 1 instead of N√σ = ∑( X - µ ) ²_________N√s = ∑( X - X ) ²_________N - 1Unbiased Sample estimatorstandard deviationPopulation standard deviation
33 Sample Standard Deviation XX - meanX - mean squared$0.00-$46.53$2,165.04$0.13-$46.40$2,152.96$0.93-$45.60$2,079.36$1.00-$45.53$2,072.98$10.00-$36.53$1,334.44$32.00-$14.53$211.12$45.53-$1.00$56.00$9.47$89.68$60.00$13.47$181.44$63.25$16.72$279.56$74.93$28.40$806.56$80.00$33.47$1,120.24$85.28$38.75$1,501.56$115.35$68.82$4,736.19$120.00$73.47$5,397.84$46.53N = 16SS = $26,295.02The standard deviation tells us that the amount of money you guys had falls an average of $41.87 dollars from the mean of $46.53Variance = $1753SD = $41.87
34 Types of Distributions Refers to the shape of the distribution.3 types:Normal distributionPositively skewed distributionNegatively skewed distribution
35 Normal DistributionNormal distributions: Specific frequency distributionBell shapedSymmetricalUnimodalMost distributions of variables found in nature (when samples are large) are normal distributions.A true normal distribution is a theoretical term – it doesn’t exist in the real world. So, we use the term “approximate” when describing our results.when the distribution of scores is very large and the scores are plotted on a line graph, the distribution tends to approximate a normal distribution.
36 Normal DistributionMean, media and mode are equal and located in the center.
38 Skewed distributions When our data are not symmetrical Positively skewed distributionNegatively skewed distributionMemory hint: skew is where the tail is; also the tail looks like a skewer and it points to the skew (either positive or negative direction)
39 Skewed DistributionsNotice how mean is pulled by the extreme scores.
40 Kurtosis Kurtosis - how flat or peaked a distribution is. Tall and skinny versus short and wideMesokurtic: normalLeptokurtic: tall and thinPlatykurtic: short and fat (squatty like a platypus!)
41 Kurtosis leptokurtic platykurtic mesokurtic Mesokurtic – have the peaks of medium heightLeptokurtic – tall and thin with only a few scores in the middlePlatykurtic – short and broadermesokurtic
42 Skewness, Number of Modes, and Kurtosis in Distribution of Housing Prices
43 z - ScoresIn which country (US vs. England) is Homer Simpson considered overweight?How can we make this comparison?Need to convert weight in pounds and kilograms to a standardized scale.Z- scores: allow for scores from different distributions to be compared under standardized conditions.The need for standardizationPutting two different variables on the same scalez-score: Transforming raw scores into standardized scoresz = (X - µ)σTell us the number of standard deviations a score is from the mean.So far we know how to describe how spread a distribution is and its shape.However, we might want to describe how an individual’s score within a distribution compares to the rest of the distribution.
44 z- Scores Class 1: M = $46.53 SD = $41.87 X = $54.76 In which class did I have more money in comparison to the distribution of the other students?Sample z-score: z = (X - M)sWhen we convert raw scores from different distributions to z-scores, these scores become part of the same z distribution and we can compare scores from different distributions.Let’ say that I asked my other class to do the same type of exercise and to provide me the amount of $ they had in their pocket on the same day that we did that in this class.Because the mean of each class is going to be different, I need to convert how much money I had to a standard measure that will allow me to compare both of my scores directly. That is, I need to convert my raw-scores from each class into the same “language” so that I can compare them properly.So first, I need to convert my raw-scores into a z-score: which is the measure of how many standard deviations my raw score is from the mean of the distribution.Class 1 : z = .20Class 2 : z = 1.94Z scores are used to transform raw scores to standard scores for the purposes of comparisons.
45 z DistributionCharacteristics: (regardless of the original distributions)z score at the mean equals 0Standard deviation equals 1
47 Standard normal distribution If a z-distribution is normal, then we refer to it as a standard normal distribution.Provides information about the proportion of scores that are higher or lower than any other score in the distribution.In a normal distribution, the z-score at the mean is 0 and the standard deviation is +1, derived theoretically.This area under the curve represents the total proportion of scores in this distribution.50% of the scores are above the mean and 50% of scores are below the mean.34% of scores fall between 0 and 1 standard deviations above the mean.47% of scores fall between 0 and 2 standard deviations above the meanBecause a normal curve is symmetrical, the same is true for scores that fall below the mean.So, 34% of scores fall between the mean and 1 SD below the mean.
48 Standard Normal Curve Table Standard normal curve table (Appendix A)Statisticians provided the proportion of scores that fall between any two z-scores.What is the percentile rank of a z score of 1?Percentile rank = proportion of scores at or below a given raw score.Ex: SAT score = 1350 M = s = 34075th percentileStatisticians have figured out the proportion of score that fall between any 2 z-scores.The first column of the table is the z-score.The second column is the proportion of scores that fall between the mean and the Z score.The third column is the proportion of scores that fall between the z-score and beyond.
49 Percentile RankThe percentage of scores that your score is higher than.89th percentile rank for heightYou are taller than 89% of the students in the class. (you are tall!)Homer Simpson: 4th percentile rank for intelligence.he is smarter than 4% of the population (or 96% of the population is smarter than Homer).GRE score: 88th percentile rankReading scores of grammar school: 18th percentile rankExamples using Standard normal curve tableFrom class data, we got the z-scores for each of your raw scores. Let use your z-scores to learn how to use this table.By saying that you did better than % of the class refers to your percentile rankTry to work backwards from percentile rank to get to a z-score.Figure out the proportion that is under the curve, see the corresponding z-score and apply to formula given the SD and mean of class data.
50 Review Data organization Descriptive statistics Z- scores Frequency distribution, bar graph, histogram and frequency polygon.Descriptive statisticsCentral tendency = middleness of a distributionMean, median and modeMeasures of variation = the spread of a distributionRange, standard deviationDistributions can be normal or skewed (positively or negatively).Z- scoresMethod of transforming raw scores into standard scores for comparisons.Normal distribution: mean z-score = 0 and standard deviation = 1Normal curve table: shows the proportions of scores below the curve for a given z-score.