Statistics Exam Notes

Introduction to Statistics

• Overview:

  • Introduction to Statistics

  • Population and Sample

  • Data and Variables

  • Levels of Measurement

  • Introduction to Add-in Data Analysis Toolpak into Excel

  • Reading: Horvath (p 1-15)

• Statistics: What’s the Point?

  • H. G. Wells (1903) hypothesized: “Statistical thinking would one day be as necessary for good citizenship as the ability to read and write”

  • Consequences of mathematical innumeracy aren't as obvious as illiteracy.

  • Lack of numerical perspective.

  • Misunderstanding of probability.

• Why Study Statistics?

  • Understanding scientific evidence requires knowledge of statistical procedures.

  • Statistics helps in evaluating arguments responsibly against those trying to influence behavior with statistical arguments.

• Science News Headline: Diet rich in animal protein is associated with a greater risk of early death

  • Journal Reference: Virtanen, et al. Dietary proteins and protein sources and risk of death: the Kuopio Ischaemic Heart Disease Risk Factor Study. The American Journal of Clinical Nutrition, 2019. DOI: 10.1093/ajcn/nqz025

  • Results:

    • Average follow-up of 22.3 years showed 1225 deaths due to disease.

    • Higher total and animal protein intakes had borderline statistically significant associations with increased mortality risk.

      • Multivariable-adjusted HR (95% CI) in the highest compared with the lowest quartile:

        • Total protein intake: 1.17 (0.99, 1.39; P across quartiles = 0.07).

        • Animal protein intake: 1.13 (0.95, 1.35; P = 0.04).

    • Higher animal-to-plant protein ratio (extreme-quartile HR = 1.23; 95% CI: 1.02, 1.49; P-trend = 0.01) and higher meat intake (extreme-quartile HR = 1.23; 95% CI: 1.04, 1.47; P = 0.01) were associated with increased mortality.

    • Association of total protein with mortality was more evident among those with a history of type 2 diabetes, cardiovascular disease, or cancer.

    • Intakes of fish, eggs, dairy, or plant protein sources were not associated with mortality.

• What is Statistics?

  • The science of the collection, organization, analysis, and interpretation of data.

  • A set of procedures and principles for collecting and organizing data and analyzing information to help people make decisions when faced with uncertainty.

• Big Picture Goal:

  • To take data from a sample and make conclusions about the population.

• Purpose of Data:

  • To get necessary information and knowledge.

  • Data + Interpretation = Information + Analysis, Discussion, Inferences = Knowledge

  • "Data" is not "information" unless it is "interpreted"

• Case Study: How Nimble Are Your Fingers?

  • Manual dexterity test: How many small pieces can you assemble in one minute?

  • Data: 200 students from a large statistics class.

  • Question: Which gender (male = 1) has better manual dexterity?

  • Summarizing the data involves measures of central tendency, variance measures, minimum and maximum scores, and the number of participants.

  • Simple summaries of data tell an interesting story and are easier to digest than large quantities of information.

  • Data are used to make a judgment or decision about a situation. This is what statistics is all about.

• The Discovery of Knowledge:

  • Asking the right question(s).

  • Collecting useful data, including deciding how much is needed.

  • Summarizing and analyzing data, with the goal of answering the question(s).

  • Making decisions and generalizations based on the observed data.

  • Turning the data and subsequent decisions into new knowledge.

Two Types of Statistics

• Descriptive Statistics:

  • Organize, describe, and summarize a small dataset.

  • Results obtained represent the entire dataset.

  • Can constitute the first stage of analysis.

  • Often used when researchers begin a new area of investigation.

  • Example: Deaths by Social Class (N=1316).

    • 1st class: SES (67% Men, 3% Women, 38% Total)

    • 2nd class: SES (92% Men, 14% Women, 59% Total)

    • 3rd class: SES (84% Men, 54% Women, 66% Children, 62% Total)

    • Total: SES (82% Men, 26% Women, 48% Children, 62% Total)

• Inferential Statistics:

  • Conclusions about populations are derived from small (random) samples.

  • Uses a smaller dataset to make estimates and draw conclusions about the greater population (that the sample is drawn from).

  • Can be used to determine cause and effect relationships, test hypotheses, and make predictions.

Population and Sample

• Population:

  • All of the objects that researchers want to describe or make inferences about.

  • Characteristic of population = “parameter”.

• Sample:

  • Sub-group of population that researcher believes represents the population.

  • A group of specific size (n=) is selected and measured.

  • Characteristic of sample = “statistic”.

  • Needs to be a good estimate of the populations parameters. Cannot measure population (too big).

  • Take a sample - a smaller group - easier to measure (e.g. 72 y10 = sample =all ages).

• Choosing a Sample from Population:

  • DO NOT DO:

    • Overrepresentation of population (e.g., 20% homeless/40% healthy/40% jobless X - sample).

    • Not a good representation of population = should include all percentages!

    • Experimental designs: manipulation.

    • Non-experimental designs: observation.

  • The best samples for experiments are those which are selected randomly!

    • Random sample means that the sample is unbiased.

    • To achieve this - every element of the population must be equally likely to be selected to the sample group.

    • Selection of one element does not affect the possibility of other elements being selected.

• Structure of Data

  • Observations (= individuals or cases).

  • Variables = observations’ attributes.

  • Data refers to any recorded observation, and are usually numeric.

  • Experimental designs = manipulation = IV to DV.

Variables

• Characteristics of a person, object, or phenomenon that is amenable to change and is measurable.

• Any observable/measurable property of organisms, objects, or events.

• Types of Variables:

  • Quantitative (Numerical).

  • Qualitative (Categorical).

• Quantitative Variables:

  • Numerical data that you can add, subtract, multiply, and divide.

  • Examples: Age (years), Blood pressure (mm of Hg), BMI, Pulse per minutes, Exercise in hours per week, Coffee drinking in ounces per day.

• Quantitative Variables: Continuous vs. Discrete

  • Continuous: can theoretically take on any value within a given range (e.g., height=188.99955… cm).

  • Discrete: can only take on certain values (e.g., no. of children in a family, No. of cities).

  • Continuous examples include a continuous spectrum variable of rainbow.

  • Discrete examples is a first ,economy , business class.

Qualitative Variables

• Binary: Two categories.

  • Examples: Dead/alive, Treatment/placebo, Disease/no disease, Exposed/Unexposed, Heads/Tails, Did you have breakfast in the morning? (Yes/No).

• More than Two categories

  • Example: Hair color – Blonde, Red-haired, Brown, and so forth.

Classification of Data (Levels of Measurement)

• Four levels of measurement:

  • Nominal

  • Ordinal

  • Interval

  • Ratio

• More information is conveyed as one moves from A to D.
  Scales of measurement can be either Qualitative OR Quantitative.

• Nominal:

  • Data placed in categories (no ordering).

  • Cannot be quantified.

  • Mutually exclusive (e.g. TYPES).

  • Examples: blood type, type of car owned, gender, colour of paint.

  • Comparative (NAMES NUMBERS among different categories).

    • KIN (400 students), PSYC (200 students), BIOL (10 students).

      • e.g. most students in which major ? -> no "average"

    • Blood type can't be combined, just compare categories.

• Ordinal:

  • Data is ranked.

  • Examples: “Idol” contest, preference (first, second, third), mineral hardness, cancer stages, University ranking, Letter-grades.

  • Used to organize data in order then nominal to categorized compare.

  • Small -> big high-low close -far Poor rich tall-short heavy light.

• Interval:

  • Equal units of measurement assigned to the attribute.

  • Zero point is arbitrary!

  • Therefore not proportional (or multiplicable).

  • e.g. temperature (F, C): temperature can be below 0 degree Celsius (-10 or -20).

  • Zero value does not mean "zero" -> it has a meaning.

  • addition and subtraction only

• Ratio:

  • Same as interval but zero is absolute or true.

  • Zero indicates an absence of the variable.

  • Therefore, direct comparison can be made.

  • Examples: Age, distance, weight, time, money etc.

  • Zero value means "zero"- nothing.

  • Numerical & means nothing = absent division / multiplication/addition/subtraction

Dependent and Independent Variables

• Dependent Variable (DV)

  • The variable of primary interest (i.e. it is measured).

  • A variable whose changes we wish to study (a response variable).

  • The variable designed to measure the effect of the variation of the independent variable (outcomes).

• Independent Variable (IV)

  • A variable we believe affects the measurements obtained on the dependent variable; i.e., it is manipulated.

  • A variable whose effects on the dependent variable we wish to study.

  • The variable that the researcher changes within a defined range, to study the effect on the dependent variable (predictors).

• Variables Example: Vitamin C study

  • Independent variable of daily vitamin C intake can determine the dependent variable of life span.

  • Scientists will manipulate the vitamin C intake in a group of 100 people: 50 people will be given a daily high dose of vitamin C and 50 people will be given a placebo pill over a period of 25 years.

  • The goal is to see if the independent variable of high vitamin C dosage affects the people's life span.

Experimental Control

• “Ideal” Scenario:

  • To imply causation, the experimenter eliminates the influence of all variables that could affect the DV except the one(s) directly manipulated.

  • All conditions are kept the same for all participants except the effect of the IV.

• “Reality” Scenario:

  • Impossible to control all variables that could affect the DV.

  • Researchers control the variables they can.

  • Other influences that are not controlled are assumed to be randomized (i.e., we assume the effects are “washed out” if they are “spread out” over the groups).

• To assist in describing data.

• In making inferences or generalizations from experimental data (sample) to larger groups (population).

• In studying causal relationships.

Introduction to Excel

• Microsoft Excel is a useful spreadsheet software.

• Use it to enter all sorts of data and perform financial, mathematical or statistical calculations.

Open an Existing Excel Workbook

• On the File tab, click Open.

Create a New Excel Workbook

• On the File tab, click New.

• Click Blank workbook.

• Excel worksheet.

Analysis ToolPak

• An Excel add-in program that provides data analysis tools for financial, statistical and engineering data analysis.

Analysis ToolPak add-in

• On the File tab, click Options.

• Under Add-ins, select Analysis ToolPak and click on the Go button.

• Check Analysis ToolPak and click on OK.

• Analysis group click on Data Analysis (Data tab).

• Dialog box appears

  • Select Histogram and click OK to create a Histogram in Excel

Organizing and Displaying Data

• How to make sense of our data?

  • Good research is based on collecting large amounts of data, which needs to be simplified.

  • Frequency distribution – lists all possible data values or type, and the frequency of occurrence of each one.

  • Meant to organize and describe the data in table form.

  • Use frequency table to construct a frequency histogram (graph).

  • Reveal the pattern of the scores/observations.

• Types of Frequency Distributions:

  • Ungrouped:

    • Frequency of all the possible data values or items in your dataset.

    • Can be nominal/ordinal categories OR quantitative but small number of single values.

  • Grouped (class intervals):

    • Applies when all “possible data values” would be too many, so data are arranged and separated into groups called class intervals.

    • Each class intervals includes a range of data.

• Types of Frequency Distributions:

  • Ungrouped: Categorical (Blood type, Majors, Teams) Quantitative: Number of kids in a household, number of town/cities you have lived in, etc

  • Grouped (class intervals): Annual salary, reaction times for any of motor tasks, weight, commuting time to York Continuous values (need a range) but can be discrete (e.g. age)

• Example:

  • Ungrouped Frequency Distribution:

    • Chin-up scores: 7, 15, 14, 9, 8, 13, 12, 15, 8, 12, 9, 9, 10, 13, 11, 10, 12 (N=17).

    • X = 15 Tally marks = II.

    • X = 14 Tally marks = I.

    • X = 13 Tally marks = II.

    • X = 12 Tally marks = III.

    • X = 11 Tally marks = I.

    • X = 10 Tally marks = II.

    • X = 9 Tally marks = III.

    • X = 8 Tally marks = II.

    • X = 7 Tally marks = I.

• Example:

  • Ungrouped Frequency Distribution w Frequency and Cumulative f.

    • X Frequency Cumulative f:

      • 15 2 17

      • 14 1 15

      • 13 2 14

      • 12 3 12

      • 11 1 9

      • 10 2 8

      • 9 3 6

      • 8 2 3

      • 7 1 1

• Example:

  • Grouped Frequency Distribution:

    • Chin-up scores (same scores as previous ex.):

      • CLASS INTERVAL FREQUENCY (f):

        • 14-15 : 3.

        • 12-13 : 5.

        • 10-11 : 3.

        • 8-9 : 5.

        • 6-7 : 1.

Steps in Constructing a Frequency Distribution

• Step 1: Count the number of scores (N = 50)

• Step 2: Identify highest and lowest score

  • MAX = 368 and MIN = 252. Range is 116.

• Step 3: Identify smallest unit of measurement

  • Smallest unit = 1.

• Step 4: Decide on appropriate number of class intervals (interval of 7; 7 rows in frequency distribution).

• Step 5: Decide on the score range of each class interval (i):

  • $i = \frac{highest \ score - lowest \ score}{number \ of \ intervals}$

  • $i = \frac{116}{7} = 16.57$

  • $i = \frac{116}{8} = 14.5$

• Step 6: Round this class interval to make this range PRETTY

  • Choose 20, 15, 12 or 10 instead of 16.57 or 14.5.

• Step 7: List class intervals of scores in order

  • Try a class interval of 15 (pretty range).

  • Need a starting value for first class interval that is also pretty.

  • Min is 252. Thus, round down to 250!

  • Class intervals (of 15): 355-369, 340-354, 325-339, 310-324, 295-309, 280-294, 265-279, 250-264.

  • Begin with the smallest values for the smallest class interval bin. Then add “i” (i.e., 15) to the next bin up until almost max.

  • Make sure that intervals have:

    • Same width (range of numbers)

    • No overlap across intervals

    • no gaps
      *Need more intervals, and thus more rows

Frequency Distributions

• Ungrouped distributions Use UNGROUPED.

  • When data are items rather than numbers, i.e., nominal or ordinal (qualitative) values.

  • When can use all possible data values without being too many (< 15) e.g. small number of possible discrete scores (e.g. how many courses this class is enrolled in this term).

  • Just list items and values and start tallying when the the number of rows needed is clear.

• Grouped distributions Used GROUPED data.

  • When data values are continuous (e.g. weight, time, blood pressure) or too many possible data values (e.g., age, or salary).

  • Calculate step 5, a range of values known as the class intervals (i) or Bins

  • A good to start is to first estimate what would be a good number of bins (step 4), but may need to redo steps 4 and 5 to get a “PRETTY” class interval or bin, e.g. 2, 5, 10, 12, 15, 20 etc (or 0.01, 0.2, 0.5 etc)
    *Add this i to the start of each bin, starting with the smallest score or value in the dataset. Cover all the data with following: (1)same width/range (2) no overlap across bins (3) no gaps (4).

Example for YOU TO DO: Construct a frequency distribution.

• Data: RTs for participants .31, .27, .28, .29, .30, .25, .26, .27, .31, .34, .27, .28, .28, .29, .32

• Steps

  • 1. N = 15 (# of scores)

  • 2. 0.34 – 0.25 = 0.09 (highest – lowest)

  • 3. 0.01 (smallest unit of measurement)

  • 4. 5 (number of categories) ß may change this afterwards

  • 5. i = 0.09/5 = 0.018 (step 2/step 4)

  • 6. 0.02 (round to “pretty number”).

• Class -Class Interval :0.34-0.35,0.32-0.33,0.30-0.31, 0.28-0.2,0.26-0.27, 0.24-0.25,

  • Count I I II III IIIII IIII I
    -Frequency 1 1 3 5 4 1

  • Cumulative Freq 1 3 13 10 5 1

• NOTE: Sometimes you will want to eliminate extreme scores from your data doing steps 2-6 to determine class intervals. But after determine the class intervals, need to add this extreme score to the tally. All scores must be included in your distribution.

Graphs:

• A pictorial representation of a frequency distribution or other data.

• Helpful in understanding concepts, e.g. frequencies, and other summary data.

• Bar graphs for Grouped data.*

• Histograms for Ungrouped data.

• Graphs for Number of Students Enrolled showing values related to faculties and school

• Graphs for Total yards offence in a session (Simple and Complex).

Histogram:

• A histogram uses vertical bars to depict frequencies of an interval/ratio variable with no spaces between the bars.

• Ungrouped FD is depicted by a bar graph while Grouped FD is depicted by a histogram.

• Histogram Examples: Age groups and frequencies

• Frequency Histogram vs Polygon (line graph)

Excel Exercise Textbook Horvath page (53-75)

• Constructing Frequency Histogram using Data Analysis Tool Example: ages of students.

• Enter the data in the excel sheet.

• Select Tools/Data Analysis on the Standard Toolbar.

• Select Histogram from the Analysis Tools window.

• Input data to construct histogram.

• Format the histogram to add title and labels.

Measure of Central Tendency

• Center of data set.

• A single summary number which indicates where many of the scores lie.

Measure of Central Tendency:

• Mean: Arithmetic average. $Mean = \frac{\Sigma x}{n}$. For example: 1.42, 1.97, 1.42, 1.50, 1.67 = 7.98 => $Mean = \frac{7.98}{5} = 1.60$.

• Median: Middle value when data is ordered. To calculate the location of the medium: $L = \frac{n+1}{2}$

• Mode: The value that occurs most often (most common value in).

Which Measure to Use?

• Nominal – Mode (Can’t calculate mean or median): e.g. Law = 64; Kine = 59; Eng. = 37.

• Ordinal – Median e.g. 1st, 2nd, 3rd, 4th, 5th.

• Interval or Ratio – Mean and/or Median

  • Use median instead of mean if highly skewed distribution or if have outliers (median not as affected).
    *Excel: Calculating central tendency measures with these equations: Mean = average(dataset) Median = median(dataset) Mode = mode(data_set).
    *Excel output must account for:
    Too many decimal places. For labs and most places, only 1-2 more decimal places than the data provided for non-integer values But for non-integer numerical answers on midterms, just to be safe, use all decimal places (paste the entire cell) OR < 10, use 2 <1, use 4 <0.01, use 5.

Graphs for Summary Data

• Distribution of data, including central tendencies like mean and medians, can also be graphically depicted

• Bar Graphs & Histograms
  Bar graphs: x-axis represents different groups or categories (nominal); and have space between bars histograms: x-axis plots the independent variable that are interval/ratio
  Graphs are created to compare relationships. Other graphs include dot plot and line plot. Key parts of the distribution come back to:

• Peaks Identify the peaks.
  Represent the most common values/bulk of data: the mode.

• Spread How much the data vary? Related to our next topic.

Kurtosis

• The relative peaked-ness or flatness of the distribution.

• It reflects whether the scores are more or less evenly distributed throughout the measurement range

• Leptokurtic The scores are bunched together with steeply sloping sides.

• Platykurtic The scores are more evenly spread out: Greater proportion of the scores fall toward the ends, or tails

Symmetry

• Normal distribution. A distribution is termed Symmetrical when the data frequencies decrease at equal rates above and below a central point *Skewed - bunching of the observations at one or the other end of the measurement range -

  • Positively Skewed: observations are bunched at the lower score values

  • Negatively Skewedobservations are bunched at the higher score values.Distributions can also be Bimodal in nature.

• The Mean is more affected than the Median if these instances occur
  Distributions with outliers should cause concern

Normal Distribution

• e.g. height, birth weight, errors in measurements or distance from bull’s eye, blood pressure, RT, MT, marks on a test (if not too easy or hard), IQ, GPA, shoe size, hours slept * For large N, e.g., N > 100

• Many variables* closely follow a normal distribution
  Normal Distribution
  Positively Skewed: income, scores on a difficult test, numbers of kids or cars or broken bones etc, points scored in a game, variables with a lower limit like weight. For large N.
  Negatively Skewed: age at death/lifespan, scores on an easy test, income as a function of age, variables with an upper limit (100%) For large N.
  *Bimodal means two peaks/mode but peaks/mode don’t need to be equal. *
  Graphs and distributions can take on all sorts of shapes.

Measures of Variability

• Measure of variability is a single number which describes the spread in a set of data.

• Example : {7, 12, 10, 8, 13} and {0, 5, 10, 15, 20} (Mean = #+ for both sets). However, the range is more spread out in the second set.

• Most common measures:

  • Range

  • Standard Deviation (SD)

Range:

• Total spread in data $Range = highest score – lowest score$

  • e.g. 4, 5, 7, 9 (Range = 9 – 4 = 5).

Interpretation – score range over which 100% of scores fall

• Advantage–very quick.

• can be used for all levels of measurement.

• Disadvantage– influenced by single extreme scores

With Examples Highlighting How Ranges Can Be Similar, Even if the Mean isn't.

Average Deviation

• How much does each score deviate (vary) from the mean?

  • e.g. 14, 12, 9, 17, 8. N=5 and Mean = $\frac{ 60}{ 5}= 12$
    *How each Score is Accounted From The Mean
    *NOTE: Add up the deviations = 0!
    *Use absolute values (i.e. ignore negative sign). Now divide by N e.g. $\frac{ 14}{ 5}= 2.8$
    This is not used because is There another method to remove the negative sign?

Variance

• Sum of Squares:

  • The sum of the squared deviations from the mean, $\,\Sigma(X - \mu)^2$

  • Always a positive value

  • Variance = divide Sum of Squares by n (statistically known as the variance)
    *Formula: $\sigma^2 = \frac{\Sigma (X - \mu)^2}{n}$
    Suppose these data represent age to the nearest year of eight persons. How would it be accounted for?
    The Spread Depends on the Deviation!

Standard Deviation

• Measure of variability for scores about the mean.

• Measure of deviations of all the scores from the mean, expressed as a single number.
  *Sample statistic, SD or s: $SD = \sqrt{\frac{\Sigma(X - \overline{X})^2}{N-1}}$

• SD-method of specifying % scores falling within certain score limits around the mean

• Quick but crude estimate: $\frac{Range}{4}$, if there are no extreme scores

Formulas

• Calculate with the Following Information:

  • Dataset A ${\ 0.5, 0.4, 0.4, 0.6, 0.5, 0.5, 0.3, 0.7, 0.6}$

  • \overline{X} = 0.50 \ ΣX2 = 2.37

  • $SD = \sqrt{\frac{\Sigma X - N\overline{X}^2}{N-1}} \ SD = \sqrt{\frac{2.37 - 9 \cdot 0.50^2}{9 - 1}} = …$

• Dataset B ${\ 0.3, 0.4, 0.7, 0.5, 0.9, 0.8, 0.2, 0.5, 0.6 }$

• $\overline{X} = 0.544 \ ΣX2 = 3.09$

  • $SD = \sqrt{\frac{\Sigma X - N\overline{X}^2}{N-1}} \ SD = \sqrt{\frac{3.09 - 9 \cdot 0.544^2 }{9 - 1}} = …$

• What can you say about the mean velocity and the variability (SD) in the 2 groups?

Dataset A${\overline{X} : = 0.50 ± 0.1225}$

• Dataset B${\overline{X} : = 0.5444 ± 0.2299}$

Interpretation of Mean and SD

Quick Reference for Means, Medians,and Distributions
*With extreme scores the mean is not a good measure of central tendency.
*Rule of Thumb: if mean and median differ by 1 SD or more then use median
Measure of Variability Across Various Methods

Nominal

Mode

Range

Ordinal

Median

Range

IntervalMean(* 𝑎𝑙𝑡ℎ𝑜𝑢𝑔ℎ 𝑚𝑒𝑑𝑖𝑎𝑛 𝑖𝑠 𝑏𝑒𝑡𝑡𝑒𝑟 𝑖𝑓 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑠𝑘𝑒𝑤𝑒𝑑)

SD

Ratio Mean (* 𝑎𝑙𝑡ℎ𝑜𝑢𝑔ℎ 𝑚𝑒𝑑𝑖𝑎𝑛 𝑖𝑠 𝑏𝑒𝑡𝑡𝑒𝑟 𝑖𝑓 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑠𝑘𝑒𝑤𝐸𝑑) SD.

Percentiles and Z-Scores

• Inter-Quartile Range (IQR):

  • IQR is a measure of variability, based on dividing a dataset into quartiles.

  • Quartiles divide a rank-ordered dataset into four equal parts.

  • Values that divide each part are called the first, second, and third quartiles and denoted by Q1, Q2, and Q3.
    Percentiles Raw scores: Percentile Raw score!

Relative Scores: Percentiles and Z-Scores

• A method of describing an individual’s standing in relation to a group, common use with norms: height/weight tables, age, fitness level, exams.

• Achieved by translating an individual’s raw score into either percentile or z-score (transformation)

• Raw scores refer to original measure e.g. height, % on exam > transform to letter grade.

• Percentiles Percentile score = the PERCENTAGE of people in the group who have the same raw score or a lower raw score, than the one in question.
  *If you have not calculated, there is a need to order the numbers and solve for where the location is
  Equation for percentile ranking {\frac{ordinal\ rank\ Of\ a\ given\ value}{ total\ # \ of\ values } *100}

Formula for Percentiles from Grouped Data -:

• $\frac{x-LL}{i} * fw + Σfb / N * 100$
  Where:
  x = the score you are converting to percentile
  LL = lower limit of the class interval that contains the score
  i = the size of each class interval
  fw = the frequency of scores in the interval that contains the score
  ∑fb = sum of the scores below the interval
  N = number of scores in the data set

Finding the Raw Score of a Given Percentile -

• Also possible to calculate in reverse: calculate what raw score a certain percentile represents
  Formula: $\ LL + P* N -[Σfb/ fwi$
  All symbols as before with the addition of