Social Stats Unit 1 Week 3

Displaying data with graphs and distribution curves


Bar graph:

  • This example is nominal data

  • Values of the variables on the x-axis

  • Frequency on the y-axis

  • Intervals have to be even/equal to one another

  • With any categorical data (nominal or ordinal) the bars cannot touch each other

  • This is a bar graph example of ordinal data

  • Difference between nominal and ordinal bar graphs: nominal has no particular order for values, while ordinal must be in order

  • Do you always have to start the y-axis at 0: not necessarily, but it is best to do that if you can/makes sense

  • Can be used to compare data (example: between 2 different cities)


Build a bar graph for nominal (NEED TO KNOW HOW TO DO ON TEST)

  1. Build and label the x-axis (values of the variable)

  2. Build and label the y-axis (frequencies)

  3. Create bars for each variable

Build a bar graph for ordinal data

  1. Build and label the x-axis (values of variable) IN ORDER

  2. Build and label the y-axis (frequencies)

  3. Create bars for each variable



Histogram:

  • Bar graph for numerical variables (interval and ratio)

  • The bars are touching because numbers flow into each other (example: group age 20-29)

  • Values of the variable go on x-axis

  • Frequency goes on y-axis


Build a histogram:

  1. Create a grouped interval frequency table with your data

  2. X-axis: create tick marks across the bottom representing the lower or upper limits of your intervals. Label tick marks

  3. Y-axis: create tick marks of equal distance for the frequencies up the y-axis. Label each tick mark. Be sure you can incorporate the highest frequency

  4. Draw the bars for each interval (touching)

  5. Label the graph with the variable name

Frequency distribution curve:

  • A distribution curve is a smooth graph that shows how often different values or ranges appear in a data set

  • X-axis: values

  • Y-axis: frequency


Interpreting frequency distributions:

  • Central location

    • Mean, Median, and/or mode

  • Spread (we’ll cover this soon)

    • Range and interquartile range

    • Standard Deviation and variance

  • Shape

    • Symmetry (evenness)

    • Modality (how many peaks)

    • Kurtosis (extreme values)


Distribution shapes

  • Normal distribution

    • Symmetrical

    • One mode

    • Mean, median, and mode are the same


Bimodal Distribution

  • Not a normal distribution (has more than one peak)


Uniform distribution

  • There is no shape/pattern to the distribution

  • Looks like a block/box


Skewed distributions

  • Look at the direction of the tail to determine skew

  • Left/negative skew

  • Right/positive skew

  • Skewed distributions and the mean:

    • When you have extreme high or low scores/data, it makes the mean an inaccurate representation of the data

      • The mean always goes towards the tail

      • If the mean is less than the median, it’s a left skew

      • If the mean is more than the median, it is a right skew

      • If they are close to each other, you have a normal curve

    • The median is the best measure of central tendency to use when you have skewed data


NEED TO KNOW MOST ABOUT SKEWED AND NORMAL DISTRIBUTIONS



Day 2

Research Q: On average, how many hours do college students sleep?

Survey Q: How many hours did you sleep last night?


Using the median for numerical data with skewed distribution

  • Symmetrical distribution (normal curve)

  • Test scores of 25 students in a stats course

  • 99, 98, 98, 97, 95, 93, 91, 89, 88, 86, 85, 85, 84, 82, 81, 80, 78, 78, 77, 77, 72, 66, 65, 64, 62

  • Mean: 82.5 Median=84


  • Asymmetrical distribution (skewed curve)

  • Test scores of 25 students in stats course

  • 99, 98, 98, 97, 95, 93, 91, 89, 88, 86, 85, 85, 84, 82, 81, 80, 78, 78, 77, 77, 72, 1,1,0,0

  • Mean: 72.6 Median: 84


How do I know if a distribution is skewed?

Pearson’s Median Skewness

  • Calculate mean minus the median

  • Multiply that difference by 3

  • Divide by the standard deviation

  • Assess the skewness

    • Greater than 1.0 (positive or negative) (ignore the negative sign ex. -1.125 is greater than 1.0)

  • Direction of skew

    • Positive number = right skew

    • Negative number = left skew



Example 1:

  • real estate prices

    • Mean: $450,000

    • Median: $380,000

    • Standard Deviation: $120,000

    • Calculation: [(450-380)*3]/120= 1.75

  • Is this data set skewed

    • Right positive skew


Example 2:

  • employee commute times

    • Mean: 28 min

    • Median: 31 min

    • Standard deviation: 8 min

    • Calculation: -1.125

  • Is this skewed:

    • Left negative skew because it is negative and it is over 1.0


Example 3:

  • Age of college students attending night school

    • Mean: 32

    • Median: 34

    • Standard deviation: 9

    • Calculation: -0.67

  • Is this skewed

    • There is no skew because 0.67 is less than 1


Stem and Leaf plot

  • Stem represents tens

  • Leaf represents ones

  • Ones have to be in perfect alignment (if turned sideways it would look like a histogram)

  • Need to have a key

    • Example: Key = 2 l 1 = 21

  • Benefits

    • preserves raw data

    • handy for small data set

    • spot outliers

    • shows the distribution shape

    • find quick statistics


Let’s create one:

Your turn:

data set onedata set two


Part 1 is the objective stuff (multiple choice/matching)

Part 2 is solving

Practice exam is almost identical to the actual exam just with different numbers

Practice exam is the best study tool