Fundamentals of Statistics

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Fundamentals of Statistics

• EBB 341

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Statistics?

• A collection of quantitative data from a sample
  or population.
• The science that deals with the collection,
  tabulation, analysis, interpretation, and
  presentation of quantitative data.

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Statistic types

• Deductive or descriptive statistics
• describe and analyze a complete data set
• Inductive statistics
• deal with a limited amount of data (sample).
• Conclusions probability?

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Population

• A population is any entire collection of people,
  animals, plants or things from which we may
  collect data.
• It is the entire group we are interested in,
  which we wish to describe or draw conclusions
  about.
• For each population there are many possible
  samples.

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Sample

• A sample is a group of units selected from a
  larger group (population).
• By studying the sample it is hoped to draw valid
  conclusions about population.
• The sample should be representative of the
  general population.
• The best way is by random sampling.

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Parameter

• A parameter is a value, usually unknown (and
  which therefore has to be estimated), used to
  represent a certain population characteristic.
• For example, the population mean is a parameter
  that is often used to indicate the average value
  of a quantity.

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Statistics
Parameters ???????2
Inferential Statistics
POPULATION
Deductive
SAMPLE
Statistics x, s, s2
Inductive
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Inferential Statistics

• Statistical Inference makes use of information
  from a sample to draw conclusions (inferences)
  about the population from which the sample was
  taken.

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Types of data

• Variables data
• quality characteristics that are measurable
  values.
• measurable and normally continuous
• may take on any value - eg. weight in kg
• Attribute data
• quality characteristics that are observed to be
  either present or absent, conforming or
  nonconforming.
• countable and normally discrete integer - eg 0,
  1, 5, 25, , but cannot 4.65

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Accurate and Precise

• Data life of light bulb 995.6 h
• The value of 995.632 h, is too accurate
  unnecessary
• Keyway spec lower limit 9.52 mm, upper limit
  9.58 mm data collected to the nearest 0.001 mm,
  and rounded to nearest 0.01 mm.

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Accurate and Precise

• Measuring instruments may not give a true reading
  because of problems due to accuracy and
  precision.
• Data 0.9532, 0.9534 0.953
• Data 0.9535, 0.9537 0.954
• If the last digit is 5 or greater, rounded up

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Describing the Data

• Graphical
• Plot or picture of a frequency distribution.
• Analytical
• Summarize data by computing a measure of central
  tendensy and dispersion.

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Sampling Methods

• Sampling methods are methods for selecting a
  sample from the population
• Simple random sampling - equal chance for each
  member of the population to be selected for the
  sample.
• Systematic sampling - the process of selecting
  every n-th member of the population arranged in a
  list.
• Stratified sample - obtained by dividing the
  population into subgroups and then randomly
  selecting from each subgroups.
• Cluster sampling - In cluster sampling groups are
  selected rather than individuals.
• Incidental or convenience sampling - Incidental
  or convenience sampling is taking an intact group
  (e.g. your own forth grade class of pupils)

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Frequency Distribution

• Consider the following set of data which are the
  high temperatures recorded for 30 consequetive
  days.
• We wish to summarize this data by creating a
  frequency distribution of the temperatures.

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To create a frequency distribution

• Identify the highest and lowest values (51 43).
  
• Create a column with variable, in this case temp.
  
• Enter the highest score at the top, and include
  all values within the range from highest score to
  lowest score.
• Create a tally column to keep track of the
  scores.
• Create a frequency column.
• At the bottom of the frequency column record the
  total frequency.

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To create a frequency distribution

• Identify the highest and lowest values (51 43).
  
• Create a column with variable, in this case temp.
  
• Enter the highest score at the top, and include
  all values within the range from highest score to
  lowest score.
• Create a tally column to keep track of the
  scores.
• Create a frequency column.
• At the bottom of the frequency column record the
  total frequency.

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Frequency Distribution
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Cummulative Frequency Distribution

• A cummulative freq distribution can be created by
  adding an additional column called "Cummulative
  Frequency."
• The cum. frequency for a given value can be
  obtained by adding the frequency for the value to
  the cummulative value for the value below the
  given value.
• For example The cum. frequency for 45 is 10
  which is the cum. frequency for 44 (6) plus the
  frequency for 45 (4).
• Finally, notice that the cum. frequency for the
  highest value should be the same as the total of
  the frequency column.

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Grouped frequency distribution

• In some cases it is necessary to group the values
  of the data to summarize the data properly.
• Eg., we wish to create a freq. distribution for
  the IQ scores of 30 pupils.
• The IQ scores in the range 73 to 139.
• To include these scores in a freq. distribution
  we would need 67 different score values (139 down
  to 73).
• This would not summarize the data very much.
• To solve this problem we would group scores
  together and create a grouped freq. distribution.
• If data has more than 20 score values, we should
  create a grouped freq. distribution by grouping
  score values together into class intervals.

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To create a grouped frequency distribution

• select an interval size (7-20 class intervals)
• create a class interval column and list each of
  the class intervals
• each interval must be the same size, they must
  not overlap, there may be no gaps within the
  range of class intervals
• create a tally column (optional)
• create a midpoint column for interval midpoints
• create a frequency column
• enter N sum value at the bottom of the
  frequency column

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Grouped frequency

• Look at the following data of high temperatures
  for 50 days.
• The highest temperature is 59 and the lowest
  temperature is 39.
• We would have 21 temperature values.
• This is greater than 20 values so we should
  create a grouped frequency distribution.

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Cumulative grouped frequency distribution
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To create a histogram from this frequency
distribution

• Arrange the values along the abscissa (horizonal
  axis) of the graph
• Create a ordinate (vertical axis) that is
  approximately three fourths the length of the
  abscissa, to contain the range of scores for the
  frequencies.
• Create the body of the histogram by drawing a bar
  or column, the length of which represents the
  frequency for each age value.
• Provide a title for the histogram.

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High temperatures for 50 days
Frequency
Temperatures
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• Histograms
• Constructing a Histogram for Discrete Data
• First, determine the frequency and relative
  frequency of each x value.
• Then mark possible x value on a horizontal scale.

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Cara Menyediakan Histogram -Grouped Data

• Tentukan nilai perbezaan, R nilai terbesar
  nilai terkecil atau R Xh - Xl
• Dapatkan bilangan turus histogram,
• Kira lebar turus, h R/t
• Nilai permulaan turus nilai terkecil data
  (h/2) atau Xl (h/2)
• Lukis histogram.

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• Histograms
• Constructing a Histogram for Continuous Data
  equal class width
• 
  Number of classes ?

Data
Relative frequency
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Bar Graph

• A bar graph is similar to a histogram except that
  the bars or columns are seperated from one
  another by a space rather than being contingent
  to one another.
• The bar graph is used to represent categorical or
  discrete data, that is data at the nominal or
  ordinal level of measurement.
• The variable levels are not continuous.

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Bar Graph
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Descriptive statistics

• Measures of Central Tendency
• Describes the center position of the data
• Mean, Median, Mode
• Measures of Dispersion
• Describes the spread of the data
• Range, Variance, Standard deviation

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Measures of central tendency Mean

• Arithmetic mean x
• where xi is one observation, ? means add up
  what follows and N is the number of observations
• So, for example, if the data are 0,2,5,9,12 the
  mean is (025912)/5 28/5 5.6

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Mean for a Population
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Mean for a Sample

• Ungrouped data
• Grouped data

n number of observed values
n sum of the frequencies h number of cells or
number observed values Xi cell midpoint
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Example - ungrouped data

• Resistance of 5 coils 3.35, 3.37, 3.28, 3.34,
  3.30 ohm.
• The average

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Example - grouped data

• Frequency Distributions of the life of 320 tires
  in 1000 km

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Measures of Location

• Central tendency

Data

• sample mean

• sample median

Provided that data is in increasing order
e.g. data 2, 2, 3, 4, 15

• Median is less sensitive to outliers.

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Median - mode

• Median the observation in the middle of
  sorted data
• Mode the most frequently occurring value

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Median and mode
100 91 85 84 75 72 72 69 65
Mode
Median
Mean 79.22
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Median

• Grouped data
• Lm lower boundary with the median
• cfm cumulative freq. all cells below Lm
• fm freq. median
• i cell interval

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Median - Grouped technique

• Use data from table above (Frequency
  Distributions of the life of 320 tires in 1000
  km).
• The halfway point (320/2 160) is reached in the
  cell with midpoint value of 37.0 and a lower
  limit of 35.6.
• The cumulative frequency is 4365163 is 154,
  the cell interval is 3, and the frequency of the
  median cell is 58
• Median 35.9 x 1000 km 35900 km.

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Measures of dispersion range

• The range is calculated by taking the maximum
  value and subtracting the minimum value.

2 4 6 8 10 12 14
Range 14 - 2 12
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Measures of dispersion variance

• Calculate the deviation from the mean for every
  observation.
• Square each deviation
• Add them up and divide by the number of
  observations

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Variance for a population

• The formula for the variance for a population
  using the deviation score method is as follows
• The mean 28/7 4
• The population variance

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Measures of dispersion standard deviation

• The standard deviation is the square root of the
  variance.
• The variance is in square units so the standard
  deviation is in the same units as x.

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Standard Deviation for a Sample

• General formula/ungrouped data
• For computation purposes

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Standard Deviation for a Sample

• Grouped data

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Example- ungrouped data

• Sample Moisture content of kraft paper are 6.7,
  6.0, 6.4, 6.4, 5.9, and 5.8 .
• Sample standard deviation, s 0.35

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Calculating the Sample Standard Deviation -
Grouped technique

• Standard deviation for a grouped sample
• Average

Table Car speeds in km/h
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Skewness

• a3 0, symmetrical
• a3 gt 0 (positive), the data are skewed to the
  right, means that long the long tail is to right
• a3 lt 0 (negative), skewed to the left, means that
  long the long tail is to left

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Kurtosis

• Leptokurtic (more peaked) distribution
• Platykurtic (flatter) distribution
• Mesokurtic (between these 2 distribution normal
  distribution.
• For example,
• if a normal distribution, mesokurtic, has a4 3,
  
• a4 gt 3 is more peaked than normal
• a4 lt 3 is less peaked than normal.

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Example
That data are skewed to the left
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Standard deviation and curve shape

• If ? is small, there is a high probability for
  getting a value close to the mean.
• If ? is large, there is a correspondingly higher
  probability for getting values further away from
  the mean.

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The Normal Curve

• The normal curve or the normal frequency
  distribution or Gaussian distribution is a
  hypothetical distribution that is widely used in
  statistical analysis.
• The characteristics of the normal curve make it
  useful in education and in the physical and
  social sciences.

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Characteristics of the Normal Curve

• The normal curve is a symmetrical distribution of
  data with an equal number of data above and below
  the midpoint of the abscissa.
• Since the distribution of data is symmetrical the
  mean, median, and mode are all at the same point
  on the abscissa.
• In other words, mean median mode.
• If we divide the distribution up into standard
  deviation units, a known proportion of data lies
  within each portion of the curve.

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• 34.13 of data lie between ? and 1? above the
  mean (?).
• 34.13 between ? and 1? below the mean.
• Approximately two-thirds (68.28 ) within 1? of
  the mean.
• 13.59 of the data lie between one and two
  standard deviations
• Finally, almost all of the data (99.74) are
  within 3? of the mean.

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The normal curve

• If x follows a bell-shaped (normal) distribution,
  then the probability that x is within
• 1 standard deviation of the mean is 68
• 2 standard deviations of the mean is 95
• 3 standard deviations of the mean is 99.7

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Standardized normal value, Z

• When a score is expressed in standard deviation
  units, it is referred to as a Z-score.
• A score that is one standard deviation above the
  mean has a Z-score of 1.
• A score that is one standard deviation below the
  mean has a Z-score of -1.
• A score that is at the mean would have a Z-score
  of 0.
• The normal curve with Z-scores along the abscissa
  looks exactly like the normal curve with standard
  deviation units along the abscissa.

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Z-value

• Deviation IQ Scores, sometimes called Wechsler IQ
  scores, are a standard score with a mean of 100
  and a standard deviation of 15.
• What percentage of the general population have
  deviation IQs lower than 85?
• So an IQ of 85 is equivalent to a z-value of 1.
• So 50 - 34.13 15.87 of the population has
  IQ scores lower than 85.

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Frequency Polygon

• A frequency polygon is what you may think of as a
  curve.
• A frequency polygon can be created with interval
  or ratio data.
• Let's create a frequency polygon with the data we
  used earlier to create a histogram.

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To create a frequency polygon

• Arrange the values along the abscissa (horizonal
  axis).
• Arrange the lowest data on the left the highest
  on the right.
• Add one value below the lowest data and one above
  the highest data.
• Create a ordinate (vertical axis).
• Arrange the frequency values along the abscissa.
• Provide a label for the ordinate (Frequency).
• Create the body of the frequency polygon by
  placing a dot for each value.
• Connect each of the dots to the next dot with a
  straight line.
• Provide a title for the frequency polygon.

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To create a frequency polygon
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